DTI.70 AD-A IpR,2 AFIT/GOR/ENS/94M-09

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1 AFIT/GOR/ENS/94M-09 AD-A S DTI.70 ELECT q 41 IpR,2 F SEVERAL MODIFIED GOODNESS-OF-FIT TESTS FOR THE CAUCHY DISTRIBUTION WITH UNKNOWN SCALE AND LOCATION PARAMETERS THESIS Bora H. ONEN First Lieutenant, TUAF AFIT/GOR/ENS/94M Approved for public release; distribution unlimited

2 AFIT/GOR/ENS/94M-09 SEVERAL MODIFIED GOODNESS-OF-FIT TESTS FOR THE CAUCHY DISTRIBUTION WITH UNKNOWN SCALE AND LOCATION PARAMETERS THESIS Presented to the Faculty of the Graduate School of Engineering of the Air Force Institute of Technology Air University Accesiorn ror In Partial Fulfillment of the NT-*1 DT,,C NTS C "AE& D T!J Requirements for the Degree of Li.. '-ou ý.d J'.: : t f,::. ~.: u Master of Science in Operations Research By. o~~~i ~.;i:/ B ora H. O N E N, B.S..... y <.,:es -'.l First Lieutenant, TUAF Oist March, 1994 Approved for public release; distribution unlimited

3 THESIS APPROVAL STUDENT: 1Lt BORA H. ONEN CLASS: GOR-94M THESIS TITLE: SEVERAL MODIFIED GOODNESS-FIT-TESTS FOR THE CAUCHY DISTRIBUTION WITH UNKNOWN LOCATION AND SCALE PARAMETERS DEFENSE DATE: February 23, 1994 COMMITTEE: NAME/DEPARTMENT SIGNATURE Advisor Dr. A. H. Moore/EN Co Advisor B. W. Woodruff/EN //I, I1J Reader Dr J. P. Cain/ENS ii

4 Preface This thesis develops powerful goodness-of-fit tests for the Cauchy distribution with the unknown location and the scale parameters. It gives some insight to relatively new techniques which are reflection or directional tests and sequential or omnibus tests. This thesis has been completed with the tremendous amount of helps coming from my advisor Albert H. Moore. I am gratefull for his knowledge, background, help and suggestions. It has been great pleasure to work with him. My thanks also go to my commitee members Dr. J. P. Cain and Maj. B. W. Woodruff for their understandings and valuable suggestions. I am and will be forever grateful to my country and Turkish Air Force for giving me such an opportunity. And my lovely classmates who never let us feel lonely and who always have been with us during the sleepless nights of AFIT, thank you all. You gave us the wonderfull chance to know some other culture and life-style. I will always remember you and AFIT which has opened a new era in my mind and life. I would like my parents Cavit and Ozden Onen for trusting me since the day I was born and providing me the best educational opportunities. I also appreciate my upper class graduates Tamer Ozmen his wife Ozlem and Erol Yiicel for their encouraging and helping us in the first year. Also my classmate Ertem Mutlu, his wife Nilgun and their lovely daughter Niler, what you have done for me is unforgetable. I also need to specially thank AFIT to give me the chance of making my most important decision, mariage, during this period. AFIT brought me luck and I met a beatiful, friendly family and their lovely daughter. This work is dedicated to my wife Oznur who has always been with me in my hard and good times. Thanks for your patience. Now it is all our time!... Bora H. ONEN i.,

5 Table of Contents Page Preface iii List of Figures vii List of Tables ix Abstract xi I. Introduction Background Problem Statement Scope Overview II. Cauchy Distribution Distribution Function Characteristic Function Properties Order Statistics Parameter Estimation Applications III. Goodness-of-Fit Tests Hypothesis Tests Goodness-of-Fit Tests Chi-squared (X 2 ) Tests EDF Tests iv

6 Page 3.3 Monte Carlo Methods Random Number Generation Random Variate Generation Inverse Transform Composition Method Acceptance-Rejection Method Bootstrap Method And Plotting Positions Parameter Estimation IV. Methodology Overview Critical Values Standard Test Reflected Test Sequential Test Power Study Power of the Standard Tests Power of the Reflected Tests Power of the Sequential Tests V. Results Critical Values Power Analysis Power Analysis of the Standard Tests Power Analysis of the Reflected Tests Power Analysis of Sequential Tests VI. Conclusion and Recommendations Conclusions Further Research v

7 Page Bibliography BIB-1 Appendix A. Computer Code For Critical Values... A-1 A.1 FORTRAN Code for Critical Values of Reflected Tests A-1 A.2 FORTRAN Code for Significance Levels of Sequential Tests A-4 Appendix B. Computer Code For Power Studies... B-1 B.1 FORTRAN Code for Power Study of Standard Tests. B-1 B.2 FORTRAN Code for Power Study of Sequential Tests. B-5 Appendix C. Probability Points... C-1 C.A Probability Points of KS and V Tests... C-1 C.2 Probability Points of CM and CM(Ref) C-12 Appendix D. Power tables of CM - V... D-1 Appendix E. Power tables of CM(Ref) - V... E-1 Appendix F. Power tables of KS - V... F-1 Vita VITA-1 vi

8 Figure List of Figures Page 2.1. Comparison of C(A = 0, i = 1) and N(p = 0,o = 1) Geometric example of the Cauchy distribution EDF and CDF Monte Carlo Study Flow Chart of Critical Value Generation For Standard Tests Flow Chart of Critical Value Generation For Reflected Tests Flow Chart of Significance Level Generation For Sequential Tests Flow Chart of Power Study For Standard Tests Flow Chart of Power Study For Reflected Tests Flow Chart of Power Study For Sequential Tests Power comparisons of CM - V against Normal Power comparisons of CM - V against Exponential Power comparisons of CM - V against Beta Power comparisons of CM - V against Gamma Power comparisons of CM - V against Weibull Power comparisons of CM(Ref) - V against Normal Power comparisons of CM(Ref) - V against Exponential Power comparisons of CM(Ref) - V against Beta Power comparisons of CM(Ref) - V against Gamma Power comparisons of CM(Ref) - V against Weibull Power comparisons of KS - V against Normal Power comparisons of KS - V against Exponential Power comparisons of KS - V against Beta vii

9 Figure Page Power comparisons of KS - V against Gamma Power comparisons of KS - V against Weibull D.1. Power comparisons of CM - V against Normal... D-12 D.2. Power comparisons of CM - V against Exponential... D-20 D.3. Power comparisons of CM - V against Beta... D-28 D.4. Power comparisons of CM - V against Gamma... D-36 D.5. Power comparisons of CM - V against Weibull... D-44 E.1. Power comparisons of CM(Ref) - V against Normal... E-7 E.2. Power comparisons of CM(Ref) - V against Exponential.... E-15 E.3. Power comparisons of CM(Ref) - V against Beta... E-23 E.4. Power comparisons of CM(Ref) - V against Gamma... E-31 E.5. Power comparisons of CM(Ref) - V against Weibull... E-39 F.I. Power comparisons of KS - V against Normal... F-7 F.2. Power comparisons of KS - V against Exponential... F-15 F.3. Power comparisons of KS - V against Beta... F-23 F.4. Power comparisons of KS - V against Gamma... F-31 F.5. Power comparisons of KS - V against Weibull... F-39 viii

10 Table List of Tables Page 5.1. Critical Values of Standard Kolmogorov-Simirnov Test Critical values of Standard Kuiper Test Comparison of different seed and plotting positions % Confidence intervals for the standard test critical values Critical Values of Reflected Kolmogorov-Simirnof Test Critical Values of Reflected Kuiper Test Significance levels of CM - V sequential test Significance levels of KS - V sequential tests Signiicance levels of KS - V sequential test Power tables of Standard Kolmogorov-Simirnov Test against alternatives Power tables of Standard Kolmogorov-Simirnov Test against t- family Power tables of Standard Kuiper Test against alternatives Power tables of Standard Kuiper Test against t-family Power tables Reflected KS and V against alternatives Power tables Reflected KS and V against t-family Power tables of CM - V against Cauchy ditribution Power tables of CM - V against Normal ditribution Power tables of CM - V against Exponential ditribution Power tables of CM - V against Beta ditribution Power tables of CM - V against Gamma ditribution Power tables of CM - V against Weibull ditribution Power tables of CM(Ref) - V against Cauchy ditribution Power tables of CM(Ref) - V against Normal ditribution ix

11 Table Page Power tables of CM(Rmf) - V against Exponential ditribution Power tables of CMkRef) - V against Beta ditribution Power tables of CM(Ref) - V against Gamma ditribution Power tables of CM(Ref) - V against Weibull ditribution Power tables of KS - V against Cauchy ditribution Power tables of KS - V against Normal ditribution Power tables of KS - V against Exponential ditribution Power tables of KS - V against Beta ditribution Power tables of KS - V against Gamma ditribution Power tables of KS - V against Weibull ditribution D.1. Power tables of CM - V against Cauchy ditribution D-2 D.2. Power tables of CM - V against Normal ditribution D-7 D.3. Power tables of CM - V against Exponential ditribution.... D-15 D.4. Power tables of CM - V against Beta ditribution... D-23 D.5. Power tables of CM - V against Gamma ditribution D-31 D.6. Power tables of CM - V against Weibull ditribution D-39 E.1. Power tables of CM(Ref) - V against Normal ditribution... E-2 E.2. Power tables of CM(Ref) - V against Exponential ditribution E-10 E.3. Power tables of CM(Ref) - V against Beta ditribution E-18 E.4. Power tables of CM(Ref) - V against Gamma ditribution... E-26 E.5. Power tables of CM(Ref) - V against Weibull ditribution... E-34 F.1. Power tables of KS - V against Normal ditribution F-2 F.2. Power tables of KS - V against Exponential ditribution F-10 F.3. Power tables of KS - V against Beta ditribution... F-18 F.4. Power tables of KS - V against Gamma ditribution F-26 F.5. Power tables of KS - V against Weibull ditribution F-34 x

12 AFIT/GOR/ENS/94M-09 Abstract Several goodness-of-fit tests such as the Kolmogorov-Simirnov and the Kuiper are studied for the Cauchy distribution with the unknown location and scale parameters. The parameters are estimated by maximum likelihood estimation. Monte Carlo simulation studies were performed to calculate the critical values for standard Kolmogorov-Simirnov and the Kuiper tests. Then a reflection technique is introduced and the critical value tables are calculated for both the Reflected Kolmogorov- Simirnov and the Reflected Kuiper tests. Several sequential tests are performed by combining standard Kolmogorov-Simirnov and Kuiper in one test, standard Cramervon Mises and the standard Kuiper in the other and finally the reflected Cramer-von Mises and the standard Kuiper in the last one. The computed critical values are then used for testing whether a set of observations follows a Cauchy distribution when the scale and location parameters are not known and to be estimated from the sample. The Monte Carlo simulations used repetitions for sample sizes of 5 through 50 with increament of 5. Throughout the study the location parameter is taken as 0 while the scale parameter is kept at 10. Power studies corresponding to each case are done and the results are presented in tables. The power studies are performed for sample sizes 5 through 50 and for a = 0.01, 0.05, 0.10, 0.15, 0.20 for the standard and the reflected tests. For sequential tests power studies have been accomplished for all of the significance level produced by combining two individual tests at form a = 0.01 to 0.20 with the increament of The Kuiper test turns out to have an overwhelming power against all distributions in standard case. The reflection technique gives an amazing improvement in the power against symmetric distributions. The reflected Kolmogorov-Simirnov has the same power as the reflected Kuiper test. Sequential tests give interesting results depending on the combination of the individual tests. xi

13 SEVERAL MODIFIED GOODNESS-OF-FIT TESTS FOR THE CAUCHY DISTRIBUTION WITH UNKNOWN SCALE AND LOCATION PARAMETERS I. Introduction Big organizations such as big factories or the Air Force always need to analyze their systems or subsystems to improve the efficiency and production level along with the quality. They try to reduce the harmful results caused by the inappropriate analysis. Since such erganizations face with complex and analytically hard problems, they need to employ statistical or simulation techniques rather than mathematical formulations. In fact, simulation and statistics are hard to separate when it comes to complex systems, because one's output usually appears to be the other's input. The basic step in the analysis of a complex system is to model the system parts. Usually the parts, the whole system or the processes can be modeled by one of the known statistical distribution functions. At this stage, some data derived out of the system serve as a reference to decide which distribution could model the system in the best way. The data is taken under some statistical processes and then the distribution which can model the system is determined. Thus, the problem becomes to test how well the sample fits to a hypothesized distribution. If a reasonable result is observed then the analysis can be carried on using that specific distribution as the model of the system part. Otherwise one could search for a better distribution. The statistical test which checks if the hypothesized distribution fits to the sample data is called goodness-of-fit (GOF) test [37:1]. Basically, GOF tests measure the 1-1

14 agreement between observed sample data and a theoretical statistical distribution. There are different types of goodness-of-fit tests and test statistics proposed so far. Among those, the most common tests are the Chi-squared (X 2 ) and the Kolmogorov- Smirnov (KS) tests. Besides these, Anderson-Darling (A 2 ) and Cramer-von Mises (CM) are the other famous GOF tests [20:382:392]. In general GOF tests are separated into two categories : (a) completely specified and (b) the modified goodness-offit tests [40:115]. In the completely specified tests, the true values of the parameters of the hypothesized distribution are known while in the modified GOF tests the parameters have to be estimated from the data. "If one foolishly used tables for the completely specified case when the parameters are estimated then the actual error is much smaller than the specified value so strongly biasing the test towards acceptance that it is almost equivalent to accepting H. without testing" [40:1151. Here the null hypothesis is Ho : The X2's are i.i.d. random variables with the distribution function F(z). Therefore, in the modified goodness-of-fit tests the parameter estimation gains importance. Although there exist many different estimation techniques, one requirement to make the GOF test tables useful is to have invariant estimators. For this reason, the method of maximum likelihood has been recommended by many statisticians. The likelihood function tells us how likely the observed sample is as a function of the possible parameter values. Maximizing the likelihood gives the parameter values for which the observed sample is most likely to have been generated, that is, the parameter values that agree most closely with the observed data [8: ]. 1.1 Background The Cauchy distribution is one of the interesting continuous distributions. Because it has no mean and variance theoretically and therefore the Central Limit Theorem is not applicable to this distribution [22: ]. The Cauchy gains its 1-2

15 importance by giving a good approximation to the normal distribution. Besides, in physics and the nuclear theory it has very wide applications [22:276]. On the other hand, the Cauchy distribution can model some economic concepts which require heavy-tailed symmetric distributions and that cannot be handled by the normal accurately [9]. These different application areas make the Cauchy distribution valuable and worthwhile to examine. Especially in the 70's, different studies on the Cauchy distribution have been accomplished mostly focusing on the proper estimation methods. The maximum likelihood estimators (MLE) were believed to have multiple roots or end up with local maximas [4]. Therefore different estimation methods were proposed. Weiss and Howlader studied on the linear estimation method for the location parameter and came up with a coefficient table [38]. Spory computed the coefficients for the best linear invariant estimation of the location and the scale parameters [34]. Koutrouvelis proposed a method for the estimation of the location and the scale parameter using empirical characteristic function [19]. Howlader and Weiss modified the Bayesian estimation and came up with the estimates comparable with MLE [15]. Higgins and Tichenor used windows estimates [13] and concluded that window estimates appear to have high efficiency for moderate and large sample sizes for specifically the Cauchy distribution and in general for the heavy-tailed distributions [14:164]. Bai and Fu proved that the MLE of location parameter converges to true parameter as opposed to the belief that the Cauchy is a possible example for the failure of maximum-likelihood method [3:140]. Haas, Bain and Antle gave iterative equations to compute MLE's [11:4041. In the literature there has been a few studies on the goodness-of-fit tests for the Cauchy distribution. One study was done by Stephens in 1990 [36]. He used weighted order statistics in estimation of the parameters. The test statistics he used for the study were Anderson Darling, Watson and Cramer-von Mises statistics. He presented a percentage point table for these statistics but didn't employ a power 1-3

16 study. Another study was accomplished by Ocasio as a master's thesis [26]. He used Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises test statistics and also presented a power study. He concluded that the Kolmogorov-Smirnov test is the most powerful among those three for any sample size [26:35]. The latest study was accomplished by Moore and Yen [23]. They applied Cramer-von Mises and Anderson- Darling tests to the Cauchy distribution and employed the reflection technique which is fairly new technique. They present the critical value tables for both cases of the tests. Moore and Yen also accomplished a power study and presented the results. 1.2 Problem Statement The powers of the previously done GOF tests for the Cauchy distribution are not too high due to the method of estimation of the parameters and the EDF statistics used. We suggest an appropriate choice of the estimation method along with the EDF statistic could result with a high power goodness-of-fit test. Specifically, it is believed that using MLEs and Kuiper statistic, a powerful test can be generated. Besides a new technique, reflection, can improve the power against symmetric distributions. On the other hand, the combination of two tests which is known as omnibus test might give some relative improvement. 1.3 Scope The purpose of the research is to apply Kolmogorov-Smirnov statistics and Kuiper to derive accompanying critical values and make a power study for the compaxison of different methods. The critical values for the KS test have already been derived by Ocasio. But this research intends to improve the accuracy. On the other hand for both tests, the new technique will be applied and the accompanying critical value tables will be derived along with the power studies. The last intention of this study is to look at couple of different sequential tests and their behavior. 1-4

17 1.4 Overevew The second chapter includes an extensive introduction of the Cauchy distribution and detailed discussion of the proposed estimation methods. Chapter 3 introduces goodness-of-fit tests, Monte Carlo analysis and some basics of Monte Carlo analysis such as random number generation. Chapter 4 gives a detailed explanation of the methodology used in this study. The results are presented in Chapter 5 as tables and graphs, and some analysis is presented. Chapter 6 lists the highlight of the results and includes some future study topics. Sample computer codes for all different tests and the power studies are presented in the Appendices. Also the complete tables and graphs of the sequential test results are presented in the Appendices. 1-5

18 II. Cauchy Distribution 2.1 Distribution Function The Cauchy distribution is a special form of the Pearson Type VII distribution [17:154]. On the other hand, it is also a member of t-family which has 1 as the degrees of freedom [22:277]. It is symmetric around the location parameter and looks like the normal except for the heavy tails. The probability density function and the cumulative distribution function are given respectively as 1 =a-(1 7ro + (9-1)2)(1 F()= + arctan( X ) (2) where \ and 4b are the location and scale parameters respectively [17:154]. 2.2 Characteristic Function The stable distributions except for the normal which is a special case can not be written explicitly. Instead, they are explained with characteristic functions [9:275]. One definition for the stable distribution is that "if X and Y axe two random variables having the same distribution function F(.), then if F(.) is stable the sum of X + Y will also have a distribution function F(.)" [9:283]. As will be explained later in this chapter, the Cauchy falls into this group. Even though there exists an explicit form for the Cauchy, the following characteristic function can be used in some cases [19:205] = 2(t) e'a%-1'p (3) 2-1

19 .3 Properdies The Cauchy distribution has some unusual properties. First of all, it does not posses any finite positive moments [17:154]. Meyer showed that the mean is indeterminate, so does not exists. The second moment is infinite, so the variance cannot be explained [22: ]. Besides there is no way to explain the skewness and kurtosis explicitly which are the functions of the third and the fourth moments respectively. As another result of having no finite mean and variance, the Cauchy doesn't have a standardized form. But usually standard form for this distribution is obtained by assigning 0 and 1 to A and 3 respectively [17:156]. Doing so, the standard pdf is 11 (4) S-.,.1 + X24 and the cdf F(z) = + arctan(z) (5) One of the main feature of the Cauchy which differs it from the others is that the Central Limit Theorem is not applicable. Kotz derived the pdf of the sum and the mean of n independent Cauchy variables using the characteristic function (3). Then the characteristic function of Sn = X'=1 Xi is So, the mean of two Cauchy variables is again Cauchy-distributed with the same A and 0b value as each Xi. And the sum has a Cauchy distribution with A = and - =!'= iki [17:156]. Therefore, the Cauchy is a stable distribution. The Cauchy distribution gives a good approximation to the normal distribution. But it has longer tails than the normal has. This is shown clearly in Figure Ai'=, 2-2

20 0.4 I,i, Mormd(0,f.)) * S 0.3 Cei -)(Q Figure 2.1 Comparison of C(A Ob, = 1) and N(' = O,,= 1) 2.1 where the same location and the scale parameters (0 and 1 respectively) were assigned to both distributions. 2.4 Order Statistics Kotz gives the summary on the order statistics of a Cauchy sample. For the Cauchy variables X 1,X 2,X 3,...,X,,, the corresponding order statistics are XI X2 _< X _<... _< X.. The probability density function of these order statistics is given by Kotz as [17:157] ii! 1 1 z- A 1 1 at z_-_ai 1i 1 x,(z) = (i - 1)!(n - i)!c2 + - arctan( )) ' 1 + _< 2-3

21 From the pdf above, the variance equation is derived as follows Var(X ) = 122 (1 - -)coec4() The expected values of the first and the last order statistics are infinite. So are the variances of the second and the second from the end. 2.5 Parameter Estimation Since estimation of paramtters have a great importance on specifying distributions, in the literature there exist a iarge amount of studies on the methods of parameter estimation for the Cauchy distribution. Unlike the most known distributions, one famous estimation technique, method of moment estimation, is not applicable to the Cauchy distribution due to the lack of finite moments. On the other hand, since it is a symmetrical distribution and gives a good approximation to the normal distribution, one could think of X9 as an estimator of A which is the location parameter. But as explained before, X has no more information than any single Xi. But instead Kotz states, "The simple form of the cumulative distribution function makes it possible to obtain simple estimators by equating population percentage points (quantiles) and the sample estimators thereof" [17:158]. Based on this idea, he derives the general formulas for the estimators of A and ik. 1 + (6) 1-(Xp - Xlp)tan(w(1 - p)) (7) where Xp is the pth percentile and p > 0.5. Kotz also concluded that the median, X 0.s, gives an unbiased estimation for A. It has become standard to pick p =.75 for the estimation of 0 [17:158]. However, 2-4

22 later studies showed that 'his method doesn't give efficient estimates, but could be used as initial estimates for some iterative methods. Another estimation method deals with order statistics. The most efficient estimators of this kind was proposed by Barnett, namely the Best Linear Unbiased Estimator (BLUE) [33:14]. Although this method requires the variances and the covariances of the order statistics be calculated first, BLUEs "... achieve full asymptotic and small sample efficiencies of 80% when compared to mle" [33:15] Later in 1977, Higgins and Tichenor proposed a new estimation technique, called windows estimates [13]. These estimates can be expressed in closed forms and are easy to compute. The study showed that window estimates have the same asymptotic distribution as MLEs and give better results for the heavy-tailed distributions, and the Cauchy distribution in particular. Comparison of windows estimates to MLE revealed that, 0 has high efficiencies for n > 10 while A has high efficiencies for n > 20 [14:164]. On the other hand, it was also shown that, for smaller sizes MLE has smaller variances than those of window estimates. This is true even for n = 40 according to the computational results that Higgins and Tichenor presented. Koutrouvelis suggests a different and simple estimation method utilizing the empirical characteristic function. The simplicity comes from the fact that, fitting the number of points t in (3) reduces the optimization for A and 0 to the determination of asymptotically optimum quantiles for the linear parameter estimation of an exponential distribution using order statistics [19:206]. The estimators appeared to be asymptotically independent and normally distributed. Koutrouvelis stated that the estimators based on this method have high efficiencies and are superior to the BLUE [19:211]. Without comparing these estimates to the MLEs, he proposes this method as an alternative to MLEs. The last estimation method covered here will be the MLE. Since MLEs became standard, every new method is compared to MLE. For the Cauchy case, this gains more importance, because MLEs for the Cauchy distribution cannot be expressed 2-5

23 in closed form. Therefore, numerical methods need to be used [11:404]. There has been questions on the convergence of the estimators due to the use of iterative method. Bai and Fu submitted a paper on the MLE for the location parameter of the Cauchy distribution [3]. They concluded that "Despite the general belief that the Cauchy distribution is an example of the failure of the maximum likelihood estimation, MLE of the location parameter converges to the true value exponentially at an optimal rate" [3:140]. Barnett on the other hand mentioned the possibility of multiple solutions due to the risk of finding local maxima instead of global maximum [4]. But Haas reported multiple solutions were never found in their study. And they concluded that the solution of the maximum likelihood equations will always be unique for distinct samples of size three or more [11:4051. Later Sours compared the MLEs with the minimum distance estimates and found that MLE gives the better, or smaller mean squared errors (MSEs) among the all minimum distance estimates [33:40]. Haas gives the likelihood function for the Cauchy sample of size n as [11:4041 LCXi,7X2,7 X3,..., Xn) = II n1 (- ~ + 1 ( A)2)) (8) taking the logarithm of (8) Log(L) = -n(log(v)) - n(log(i7)) - n log(1 + ( A)2) (9) 1=1 Then taking the partial derivatives of (9) and setting them equal to 0 gives the following maximum likelihood equations = I - 0 (10) "n 1 1 (11) = 1 + (i--)

24 The Princeton study used an iterative method derived from (10) and (11) [2:2C3-17]. The iterative equations in which there is a need for initial values for both A and are shown below,=(x- ) (12) 1= 2+(Xi-;k&)3-203/ 2 i=1 n 2 # +(x. -.X (13) For this iteration method, the initial value for A is chosen as the sample median and assigned as A 0. For the scale parameter -0, the semiquantile distance is picked as initial value iko. Semiquantile distance is obtained from (7) by assigning p = Applications In 1970's, it has been realized that in the economic modeling some data are flatly inconsistent with the hypothesis of normality. The observed data had much weight in the extreme tails. Then the Cauchy distribution and the other stable distributions were assumed to give better fit to the data. It has been observed that this holds true for time series analysis, stock and commodity price changes, sales, employment or asset size measures of business firms and personal incomes [9:275]. Another application of the Cauchy distribution is closely related to the normal distribution. Suppose Y and V are normally distributed as N(0, 1). Then the variable Z wbich is the ratio of Y to Z has a Cauchy distribution [17:160]. Meyer explains a physical situation from which the Cauchy distribution may be obtained in the real world [22:276]. Consider we mounted a machine gun at a unit distance from a wall. Figure 2.2 shows the direction of the gun. Then the gun is rotated at a constant angular velocity, d = w, and is fired at a constant rate. A hit occurs for <_ 0 <_ d where 0 is uniformly distributed over the range - < 0 < E Then f(b)

25 0 z-axis i/ Figure 2.2 Geometric example of the Cauchy distribution for the defined range. Since the distance from the wall it; unity, tan(z) = 0 = arctan(z ) = O( W) and The derivative of O(z) is do 1 dz 1 + X2 The pdf of x is then computed as q(z) dc f (O(z)) IIdz- 1 1 W-l+X2 which is exactly the standard Cauchy pdf as in (4). Meyer also states that the Cauchy distribution arises in the theory of atomic and nuclear transitions [22:276c. Kotz gives some walls unty the Brownian motion which tend to a Cauchy distribution [17:161]. 2-8

26 III. Goodness-of-Fit Tests 3.1 Hypothesis Tests Statistical analysis includes hypothesis tests prior to the analysis. In hypothesis testing, first a claim believed to be true is made. Then a random sample ir drawn from the population and a decision is made for or against the hypothesis. This procedure includes following steps [21: ]: 1. A hypothesis to be tested and believed to be true is made. This hypothesis called null hypothesis and denoted as Ho. 2. The negative of the null hypothesis is set up and called alternative hypothesis (HG). 3. A test statistic is chosen. The test statistic is "a function of the sample data on which the decision (reject Ho or do not reject) is to be based. 4. A rule which is related to the rejection region is established to make the decision. The rejection region specifies the values of the test statistic for which the null hypothesis is rejected. These cutoff values of the set statistics are called critical values. Then the decision rule is "* reject H,, if the value of the test statistic computed from the sample is greater than the critical value "* accept Ho, if the value of the test statistic is not in the rejection region Hypothesis testing is based on the sample data. However, the sample cannot carry all the information about the population. Therefore there exists a possibility of making errors in decision making, which can occur in two types: * Type I error occurs if Ho is rejected when it is actually true. This error is denoted as a. 3-1

27 * 7ype II error is made if H, is accepted when actually H. is true. Type II error is denoted as 3 [21:430]. The hypothesis testing is concerned minimizing these two types of errors. The maximum probabilities of making type I error have been given the labels of a, which is called significance levels. The hypothesis tests are done based on the a levels. The maximum probabilities of making type II error which is labeled as '3 is used in the determining the power of the test. (1 -/0) which denotes the probability of rejecting Ho when H. is true, gives the power of the hypothesis test. 3.2 Goodness-of-Fit Tests Goodness-of-fit (GOF) tests are used to examine how well a sample of data fits to a hypothesized distribution. In fact, GOF te:ts are regular hypothesis testing in which the null hypothesis, Ho, is that the data comes from the hypothesized distribution F(z). GOF tests can be separated into two subgroups as graphical and using test statistics. Following sections will explain the tests using test statistics Chi-squared (x 2 ) Tests. The first test introduced by Pearson in 1900 is the Chi-squared test. The basic idea of the chi-squared tests is to reduce the general fitting test to a test based on comparison of observed cell counts with their expected values under the hypothesis to be tested [37:63]. Although the chi-squared tests are the most generally applicable tests, they are often less powerful than the other tests due to the decrease in information caused by the grouping of the data [40:113]. The general concept of the chi-squared test can be summarized as in the following paragraph. Suppose we have a random sample X 1, X 2,..., X,, with the distribution F(a). Pearson partitioned the range into n cells. Oi's are the observed number of Xis in the it' cell. Then Oi has a binomial distribution and therefore, np gives the expected 3-2

28 value of O0 which is the number of Xis that should fall in the ith cell theoretically. Pearson reasoned that the difference between the observed and the expected cell frequencies, Oi - np,, expresses lack of fit of the data to F(z). He suggested the chi-squared test statistic as the function of this difference. h (0, - E,) 2 SE, is chi-squared distributed with the degrees of freedom k - p - 1 where p stands for the number of parameter estimated, k the number of cells and Ei = npi [24:64-65]. The test results with rejection if i2 > XL-2, where X -2-_ refers to the critical chi-squared value. Another draw back of the chi-squared tests besides the low power is that it is subjective. Because the choice of the number of cells is arbitrary with the limitation of having at least 4 observations at each cell. Therefore, the result of the test is not unique and it may change with the choice of cell numbers. It is recommended to use samples of size greater than 25 for the X 2 test [40:114] EDF Tests. The second group of GOF test statistics are EDF statistics. "Empirical distribution function (EDF) is a step function, calculated from the sample, which estimates the population distribution function" [37:97]. With Stephens' words "EDF statistics are measures of the discrepancy between the EDF and a given distribution function, and are used for testing the fit of the sample to the distribution... " [37:97]. For a sample of size n which is X 1, X 2,..., X,. from the distribution of F(x), the EDF (F, (z)) is defined as number of observations < x n 3-3

29 For ordered statistics, EDF is specifically defined as F,() = 0, x < X(i) F.(x) = 1, X(i) _<zx< X(i+,), i= l1,...,n- I1 Fn(z) = 1, X(,,) < 3 The difference between CDF and EDF is shown at Figure EDF CDF Figure 3.1 EDF and CDF EDF statistics are separated into two major groups from which the most common EDF statistics are drawn. The first group is quadratic statistics which are also called the Cramer-von Mises family. These statistics are generated from Q = nj {F0(x) - F(X)I2)(x)dF(x) (14) 3-4

30 where 1P(m) is the suitable function which gives weights to the &F,,(z)-F(z)} 2. When 41(z) = 1, Cramer-von Misses statistics (CM) is obtained. 12(z) = 1/{F(X)(1 - F(z)} gives the Anderson Darling statistic (A 2 ) [37:100]. The second group of statistics are called supremum statistics. The basic statistics of this kind is D+ and D- which are the largest vertical difference when F(z) is greater than F(z) and smaller than F(z) respectively. They are defined as D+ = max(f,(z) - F(z)) D- = max(f(z) - F&(z)) The most common EDF statistic, Kolmogorov-Smirnov statistic (KS) is a function of these two basic statistics. Precisely, KS = max(d+, D-) The other EDF statistic, Kuiper statistic (V) is also a function of D+ and D-. It is defined as V=D+ +D- KS and V are the test statistics that are developed in this thesis. Stephens gives the computational formulas for these EDF statistics along with the short discussion. According to his explanation, for any distribution of F(z), F(zi) is uniformly distributed. Therefore, computing the EDF statistics comparing the EDF of F(zi) with the uniform distribution is the same as comparing the EDF of zi with F(z) [37:101]. This conclusion leads to the following practical computational formulas of the previously defined EDF statistics: D+ = max(- - F(z 1 )) (15) 3-5

31 D- = mx(f - (- )) (16) Kolmogorov-Smirnov and Kuiper statistics are computed from (15) and (16) as KS = max(d+,d-) (17) V = D+ + D- (18) The Cramer-von Mises and the Anderson-Darling statistics are modified from (14) and turn out to be "M -2i n (19) A -n - - (2i - 1)[n F(xi) + In (1 - n i=1 The EDF tests can be used with small samples, unlike the chi-squared tests. One the other hand, the EDF tests could only be used when F(x) was fully specified, that is, the parameters were known. Because with a fully specified CDF, the probability integral transformation converts CDF values to ordered values in the interval of [0, 11 based on a uniform distribution. If the parameters of F(x) were to be estimated, the CDF of EDF statistics would depend on the sample size and the value of the parameters. This prevented the widespread usage of the EDF statistics [35: ]. David and Johnson, in 1948, showed that if the parameters to be estimated from the sample are the invariant estimators of only location and the scale parameters, then the CDF of EDF statistics will depend on the functional form of F(z), not on the estimated parameters [7]. This clarification made the modified GOF tests to be more widely used. The modified GOF tests are the GOF tests in which F(x) is not specified and the parameters are estimated. In the literature there are numbers of studies on the goodness-of-fit tests each of which uses different estimation techniques, different test statistics, different methods 3-6

32 in calculating critical values and is done for different distributions. Daniel prepared a bibliography on the GOF studies in 1980 [6]. The bibliography goes back to 1900 when Pearson first introduced the X 2 test. Then it covers the all major studies till The most studied distributions appeared to be the normal and exponential distributions. There are various kinds of methods and tests studied along with the studies on the efficiencies of the tests and the asymptotic theories of the test statistics. Besides tho se studies, there exist three important resources in the literature on the GOF. The first one is the Goodness-of-Fit Techniques written by Stephens and D'agostino [37]. They refer to numerous studies and present various kind of GOF tests giving examples on different distributions. The second book is Smooth Goodness of Fit Tests written by Rayner and Best [27]. The third resource is on the multivariate data analysis which is Goodness-of-Fit Statistics For Discrete Multivariate Data written by Read and Cressie [28]. The majority of these studies intended to develop a new technique or modify the ones already proposed to increase the power. While some studies are searching for new estimation technique for this reason, some are modifying the EDF statistics with different plotting positions. Besides the standard GOF tests, a new technique, directional test, was proposed. The idea of the directional test also known as the reflected test has been motivated from Schuster's papers [31]-[32]. In the first paper Schuster derives a second sample from the original center. Then he uses the sum of the CDFs of the two samples and derives a new Kolmogorov-Smirnov kind statistic. In the second paper, he proposes a new method of parameter estimation using the same method of deriving the second sample. In short, he estimates the location parameter and then reflects the sample around the location parameter. Thus he gets the asymmetric of the original sample. This concept was modified by Ream and used in the GOF tests for normality. He basically derived the second sample as Schuster did. But instead 3-7

33 of treating the second sample itself, Ream combined the two samples and dealt with the new sample of doubled size. The reason for the reflection was to improve the power increasing the sample size. The results turn out as expected for the symmetric distributions. But "no improvement would be evident in powers generated against the non-symmetric distributions" [29:61]. Sequential tests, also called omnibus tests, are based on the idea that two different tests are run independently. The order of the tests is not important. But the test of running two independent tests has its own significance level and power. The significance level of the sequential test which is the combination of the test1 at a, and test2 at a 2 is a <_ al + a 2 (20) Pearson, D'Agostino, and Bowmen showed that using both the skewness and the kurtosis tests for normality if a, = a2 =- a* then an approximation to the overall significance level is [37:390] a = 4(a* - (a*) 2 This idea can be applied to EDF statistics using two of them at the same time. The new test would have a separate significance level and different power. This study will introduce three new sequential tests for the Cauchy distribution. Stephens presented a report on the EDF tests for the Cauchy distribution. He derived the percentage points for the CM and A 2 and also U 2 (Watson statistic) [36]. Although the report focuses on these statistics from beginning to the end also the percentage points were found for KS and V statistics. The report concludes that the EDF statistics give higher power in the case of CM and V but no values from the power study is presented. Stephens uses a different method for the estimation reasoning that the MLEs are hard to work with. On the other hand, he states that the asymptotic theory is not applicable to these statistics, he doesn't mention how he calculated the percentage values. 3-8

34 The other GOF study for the Cauchy distribution was achieved by Ocasio in 1985 [26J. He derived the critical values for the CM, A 2 and KS statistics for the Cauchy distribution with unknown shape and location parameters. He used MLEs and bootstrap method with 5000 samples and also did a power analysis against various distributions. The study concluded that the KS test is the most powerful one among those three tests for the sample size n greater than 5. The last study was done by Moore and Yen on the CM and A 2 tests [231. They used MLEs with 5000 samples and also included the reflection technique for both of the tests. Moore and Yen present the critical values along with the power tables. The study results support the idea that the reflection improves the power against the symmetric distributions. 3.3 Monte Carlo Methods Some systems are mathematically difficult systems, and hard to define in straight forward closed form equations. When an exact mathematical model can not be developed economically or when it becomes too complex to permit timely evaluation, such complex systems are usually analyzed using Monte Carlo Method. Law and Kelton define Monte Carlo simulation as "a scheme employing random numbers, that is, U(0, 1) random variates, which is used for solving certain stochastic or deterministic problems where the passage of time plays no substantive role" [5:113]. Even though it just says U(0, 1) random variates, usually different random variates are used but as explained in the latter sections they are all generated from U(0, 1) random variates. The common usage of the Monte Carlo analysis is in the reliability analysis. The second area is the deterministic problems. Since it is a simulation process, it has very large application area. The only difference of it from the simulation is that it doesn't deal with time. 3-9

35 Monte Carlo simulation is now widely used to solve certain problems in statistics that are not tractable. For example, it has been applied to estimate the critical values or the power of a new hypothesis test. Determining the critical values for the Kolnogorov-Smirnov test for normality... is such an application. [5:114] The statisticians have come up with asymptotic distributions of the EDF statistics. However, they are still difficult to estimate and work with. Therefore many of the goodness-of-fit studies employ Monte Carlo methods. Noree states that In general, a valid Monte Carlo significance level can be computed for any test statistic that is a function of data drawn from any specified population. The population does not have to have a familiar, well-behaved distribution studied by statisticians; the population can be entirely arbitrary. [25:491 The Monte Carlo process usually involves the determination of the distribution of interest (mostly related to the element in the system), selection of the random sample from this distribution, combining of these samples to obtain the measure or information required. "The process of random selection and determination of the system effects are repeated a large number of times and, each repetition results in another different estimate of the system characteristic that is being measured" [18:3-1]. The accuracy and reliability of the Monte Carlo method are based on the law of large numbers which is stated as, as the sample size gets bigger, the difference between the sample mean and the population mean becomes smaller [1:176]. For a big enough sample size, the sample mean is equal to the population mean. However, big enough is a relative term. It has been shown that for the normal distribution, samples of n > 30 is enough to assume sample mean is equal to the population mean. But there doesn't exist any comment for the other distributions. However, most of the Monte Carlo studies use 5000 iterations. Thus, 5000 independent data are obtained and used as representative of the system. 3-10

36 The weakness of the Monte Carlo method is that the uncertainty of the raw data. But Gwinn states the opinion of Hammersley and Handscomb on this uncertainty Good experimentation tries to ensure that the sample shall be more rather than less representative... [Monte Carlo results] can nevertheless serve a useful purpose if we can manage to make the uncertainty fairly negligible, that is to say to make it unlikely that the answers are wrong by very much. [10:2-14] Then the uncertainty can be made negligible by increasing the number of observations or in other words number of replications. The Monte Carlo study can be generalized for the purpose of this thesis as shown in Figure 3.2. Next sections will include different methods for these main steps of the Monte Carlo analysis. 3.4 Random Number Generation Almost all simulation processes require random samples or deviates. In nature, there is no such a thing as a random number. But there exist various arithmetic procedures to generate random numbers. Since the procedures employ deterministic rules, they are called pseudo-random numbers, meaning supposedly random but not really. Ripley defines random numbers as "A sequence of pseudo-random numbers (Ui) is a deterministic sequence of numbers in (0,1) having the same relevant statistical properties as a sequence of random number" [30:15]. All random variates from different distributions are generated using uniform, U(0, 1), random numbers. Thus, the generation of the uniform random numbers gains the most importance and is the basics of random number generation. Therefore this section gives an introduction to major methods of generating U(0, 1) random numbers. The earliest methods were so crude, and carried out by hand such as throwing dice, dealing out cards, casting lots or drawing numbered balls from an urn. Later, 3-11

37 n -- Determine the Distribution n repeti n Generate n Random Deviates S Estimate the Parameters -- Derive Necessary Information 1 Estimate the System Chareteristic STOP Figure 3.2 Monte Carlo Study 3-12

38 some electrical devices were developed just for generating random numbers. As the computer got widely used, numerical or arithmetical methods which are based on the computer operation system were generated. The first of this type was midsquare method proposed by von Neuman and Metropolis. But "One serious problem (among others) is that it has a strong tendency to degenerate fairly rapidly to zero, where it will stay forever" [20:422]. In 1951, Linear Congruential Generators (LCGs) were introduced by Lehner. The LCGs have the form of Z= (azi + c)(mod m) where m, a, c, and Zo are nonnegative integers and m > 0, m > a, m > c, Zo < m. LCGs have a looping behavior, that is, the same sequence of random numbers will repeat itself whenever Zi = Z 0. This length of cycle is called period and when it is equal to m, it is called full period. But to make the sequence full period there are some other requirements as explained by Law and Kelton [20:426]. If c > 0 it is called mized LCG. Mixed LCGs have some advantages the most important of which is that if m = 2 b where b denotes the number of bits in a word on the computer it helps "... to avoid explicit division by m on most computers by taking the advantage of integer overflow" [20:4271. When c = 0, the generator is called multiplicative generator. Multiplicative LCGs have the advantage of not having the addition of c, but the disadvantage of not being able to have full period [20:429]. In general, LCGs are the most commonly used generators. However there exist several different generators such as composite and Tausworthe generators. But the facilities used in this study use LCGs. Therefore the other methods will not be explained. But the general information about the other generators can be found in Law and Kelton's book [20]. 3-13

39 3.5 Random Variate Generation There are many different techniques for generating random variates. The choice of the technique depends on the type of distribution from which the variate will be generated. The techniques can be classified into several general groups. The following sections will discuss these general groups Inverse Transform. Suppose the variable X has the CDF F(z), then F(m) = u has the inverse CDF denoted as F(u)- 1 where u is uniformly distributed. Therefore to generate random variate from the distribution function F(z) the following algorithm is used: 1. Generate U -, U(0, 1) 2. Calculate X so that X = F(U)-1 The inverse transform method can be applied to the continuous, discrete and the mixed distributions. But one disadvantage of this method is that there may not be a closed form formula of the CDF as in the normal and gamma distributi3ns. On the other hand, for some specific distributions the method may not be the fastest method [20:4721. Despite these drawbacks, one important advantage is that the method can facilitate variance-reduction techniques such as antithetic variates. The second advantage is that it is easy to generate from truncated distributions. The final advantage of the inverse-transform method is that generating order statistics is very easy with this method Composition Method. This technique is used when it is possible to explain the CDF of the distribution from which the variate will be generated as a convex combination of the other CDF's such as F(z) = k j=1 3-14

40 where k Epi = 1 and pi > 0 j=1 which is the convexity constraint. The algorithm is given as 1. Generate a random integer Je{1, 2,..., k} such that P(J = j) for j = 1,2,...,k 2. Generate random variate X from the distribution with CDF Fj. [20:474] method. The composition method is faster, in some cases, than the inverse-transform Acceptance-Rejection Method. This method is not a direct method as the other methods and can be useful when the direct methods fail or are inefficient. The idea of the acceptance-rejection method depends on the idea that a function t(z) can be defined such that t(z) majorizes the density f(z). This requires t(x) Ž f(_) for all x. c f 00 t(z)dz > fj00f(z)dx -=1 Dividing t(x) by c gives the density function r(z) = Thus since CDF of r(z) will be uniformly distributed, it is possible to generate variate from r(x). Then the algorithm is given by 1. Generate Y from the majorizing PDF r(z). 2. Generate U - U(0, 1) independent of Y. 3. If U < f(z)/t(y), then accept X = Y as the variate from f(z). Otherwise reject the value and go back to step 1. [20:478] 3-15

41 The important step in the acceptance-rejection method is to choose t(z) properly. The majorizing t(z) should be picked so that the generation from r(z) would be easy, and c is small, that is, t(z) should fit closely above f (z). The first requirement is to increase the speed while the second is to increase the accuracy [20:479] The techniques are modified to generate from specific distributions. But these techniques are the general forms of the random variate generation. 3.6 Bootstrap Method And Plotting Positions The plotting positions methods is the most common method to derive the critical values for a GOF test. It depends on the bootstrap method. The bootstrap methods were pioneered by Efron for estimating confidence intervals. But they can be modified to estimate the significance levels. There exist many bootstrap methods which use different modifications. The basics of this technique are explained below. Suppose z is a random sample and t(x) is the value of test statistic of hypothetical x. Since x is a random variable, t(z) is a random variable too, with its own probability function. Then P(t(z) _Ž h) gives the sampling distribution of t(z). Now let xo be the real random sample from the real population; t(z 0 ) is the value of the test statistic for that real sample. A hypothesis test consists of calculating how unusual t(zo) is relative to the sampling distribution of t(x). That is, significance of the test statistic ideally is prob(t(x) Ž_ t(xo)) and the rule for rejecting the null hypothesis is Reject if prob(t(x) Ž t(zo)) < a The problem in assessing a significance level thus reduces to estimating the sampling distribution of the test statistic under the null hypothesis, i.e. the probability distribution of t(z)... The sampling distribution is estimated by drawing simulated random samples from the null hypothesis population. The significance level is essentially the proportion of simulated samples for which the value of the test statistic was at least as large as for the original sample. [25:64] 3-16

42 The procedure explained above is an application of the Monte Carlo to draw random sample. "In fact, given a sample from a population, the nonparametric maximum likelihood estimate of the population distribution is the sample itself" [25:65]. Therefore, procedures for deriving the critical levels can be applied by sampling with replacement from the sample. Then plotting position technique is employed to derive the critical values. It has been shown that this technique is more precise than that to select the order statistic which, as a percentage of the total statistics, matches the percentile level. Plotting position method is accomplished by approximating a piecewise linear function for the discrete order statistics. Then, it would be possible to interpolate between the discrete values of the statistics and get more accurate critical values. The interpolation is done plotting the order statistics against a plotting position which represents the order statistics on a 0 to 1 scale. The method of the interpolation is explained in the next chapter. Many different plotting positions have been stated so far. Some of them have been used in different goodness of fit studies. The most famous one is called the mean plotting position and computed by (i - 0.5)/n where i is the rank of the order statistic and n is the sample size. Some of the other plotting positions are the median rank (i - 0.3)/(n - 0.4), mode (i - 1)/(n- 1) plotting positions. Different plotting position methods arise from the need of plotting ordered data against the CDF value. The CDF is a step function that jumps from (i - 1)/n to i/n at the ith order statistic of the sample. If (i - 1)/n is used as plotting positions then the smallest order statistic can not be plotted, while in the case of i/n the largest statistic is not possible to be plotted [12: There exist many studies on the plotting positions. Among those, Harter [12] published an extensive analysis of the different plotting positions proposed. The studies meet at the same objective which is to look for plotting positions that produce minimum variance unbiased estimates or minimum mean square deviation 3-17

43 of a biased estimate. Harter concluded that the median plotting position yields median unbiased estimates and "One may wish to avoid the difficulties associated with unbiased estimates by obtaining median unbiased estimates instead" [12:1625]. For the sample size smaller than 20, different plotting positions may give better results. But for the sample sizes over 20, the difference between the various plotting positions are insignificant. 3.7 Parameter Estimation Any kind of statistical analysis based on a sample improves its accuracy and efficiency by first employing the best estimation method. The Monte Carlo study is no exception to this rule, especially when used in goodness-of-fit tests. Chapter 2 discussed different estimation methods proposed for the Cauchy distribution. Among those, the MLEs have been selected as the most appropriate estimators for this study. One important reason for this is that, different studies concluded that MLEs have smaller variance or MSE than most of the other estimators of the Cauchy distribution. Besides, it has been proved that if there exists a sufficient estimator of a parameter, the MLE is definitely based on this sufficient statistic. Also, "maximumlikelihood estimators posses certain desirable large-sample properties" [39:345]. The most important property of MLEs for this study is that they are invariant. That is, If j is the MLE of 0 and h(o) is an inverse function, then h(d) is the MLE of h(o). By inverse function it is meant that there exists one-to-one relationship between values of 0 and the corresponding values of h(o) [39:349]. Thus, empirical distribution functions and the test statistics used in the study become the MLEs of the real values with the desired properties. 3-18

44 IV. Methodology 4.1 Ov~erview The distribution of the EDF statistics have been studied for years. But, statisticians couldn't come up with nice, easy-to-apply formulations. As mentioned before, asymptotic distribution of the Kuiper statistic has been studied too. But, it is a general agreement that any closed form of the distribution functions of EDF statistics is hard to deal with. As a result of this common belief, the Monte Carlo analysis was referred to derive information about the EDF statistics. This thesis examines three types of goodness-of-fit tests. The first one is the standard test. The standard test was applied to both the Kolmogorov-Smirnov (KS) and the Kuiper (V) test statistics, and therefore the critical values were computed for them. The second type is the reflected, or directional test. This method was used again for both KS and V statistics and the critical values were generated. The third type of goodness-of-fit tests in the scope of this theses is the sequential test. This type combines two different tests. The sequential test was applied to three different combinations. The pairs are standard CM and standard Kuiper, reflected CM and standard Kuiper, standard KS and standard Kuiper. Even though the critical values for both cases of CM test for the Cauchy distribution were generated by Moore and Dr. Yen, they were regenerated using the same parameters and the methods as in the other tests. The Cauchy samples used in all the critical value computation were arbitrarily picked from C(0, 10). After the critical values were computed for all of the tests, some power analyses were done using different alternative distributions. All the codes for either critical value computations or the power studies were written in FORTRAN 77. To reduce the running time, IMSL STAT/LIBRARY subroutines were widely used [16]. All the codes were run on Sparc station 2 machines. For the reasons mentioned before, MLEs were selected as the estimators used in this thesis. The computer code for the MLEs is a modified version of the FORTRAN 4-1

45 code CMLE used in the Princeton study [2]. This subroutine basically uses the equations (12) and (13) and solves them iteratively. Sours showed that 100 iterations is enough for convergence [33]. The convergence is determined within c = for the location and e = 0.05 for the scale parameter. But the way of computing the median was coded just for the even sample sizes in that study. To improve the accuracy, the code was modified in this research and added another part for the odd sample sizes. On the other hand, Princeton study used the modified semi-quantile as in (7) for the initial estimate of scale parameter. In this study, since the true value of 0& was known as 10, it has been used as the initial estimate with the idea that it might increase the accuracy and reduce the computational time. The previous goodness-of-fit studies using Monte Carlo method implemented, in general, 5000 independent values of test statistics. Since the sample size is a lot bigger compared to 20, any one of the plotting positions is justified. For this thesis, to get more accurate results, 50,000 iterations were used and 50,000 independent values of each test statistic have been generated. Therefore, it is intuitive that any kind of plotting position method could have been good to find the critical values. But as a choice, the median rank plotting position method has been selected for the purpose of this thesis. The following sections will introduce the methods in detail for each type of the tests and the power studies. 4.2 Critical Values Standard Test. The Monte Carlo procedure as explained earlier has been modified to generate critical values. The detailed flow chart of the generation process is shown on the Figure 4.1. This thesis includes three different test types namely standard, reflected, and sequential. The standard and the reflected tests use the same procedure to generate 4-2

46 :START 50,0001 p igenerate n Random Deviates from Cauchy(0,10) [ Order the Variates Calculate the MLEs for Scale and Location Parameters E Calculate the Hypothesized CDF Values ] Calculate the EDF Statistics Order the Statistics Calculate the Critical Values (STOP) Figure 4.1 Flow Chart of Critical Value Generation For Standard Tests 4-3

47 critical values except for reflection. So this procedure will be explained here alone and then the reflection part will be discussed. 1. Step I : Random deviate generation. For each run of the Monte Carlo study a specified size of Cauchy sample is needed. This has been accomplished by using the IMSL subroutine RNCHY. This subroutine generates a sample from the Cauchy distribution with A = 0 and 4) = 1 using the inverse transformation method as explained earlier. To produce a sample with different A and 0 value, the deviates are multiplied by the new 4' and A is added [16:997]. This thesis uses samples from C(0, 10). So, each deviate was just multiplied by 10 after the generation. 2. Step 2 : Parameter estimation. As mentioned earlier CMLE subroutine of Princeton study was used to estimate the location and the scale parameters [2]. This subroutine requires the sample to be ordered first. Therefore, after the sample was generated the deviates were sorted in ascending order and then the CMLE calculated the MLEs. 3. Step 3 : Calculate the CDF values. This was done by substituting the computed MLEs in the CDF (2). Then the hypothesized CDF value for the each deviate was computed. 4. Step 4 : Compute the test statistics. The code to compute the KS test statistics was modified from Sours code [33:68-69]. KS was found using the equation (17) as the maz(d+, D-). Therefore, first D+ and D- values were computed as in (15) and (16) respectively. Finally the max of these two maximums was picked as the KS test statistic. The Kuiper test statistics (V) were computed by adding the already computed D+ and D- together. 5. Step 5 : Repeat the steps (1-5) 50,000 times to generate 50,000 independent test statistics. 4-4

48 6. Step 6 : Order the statistics. To assign the plotting positions to the test statistics, the statistics must be in ascending order. ordered using a simple code. Therefore, they were 7. Step 7: Determine the critical values. This technique depends on the bootstrap method. This method is accomplished by approximating a piecewise linear function for the discrete order statistics. Then, it would be possible to interpolate between the discrete values of the statistics and get more accurate critical values. "The interpolation is done plotting the order statistics against a plotting position which represents the order statistics on a zero to one scale" [26:24]. The plotting positions used in this thesis are the median rank plotting positions which is computed as (i - 0.3)/(n - 0.4). The basic idea in this process is that the critical value at a certain a level is the (1 - a)th percentile. So, to find the percentile corresponding to a certain a level, the greatest plotting position less than that percentile is found. Using this value and next to that, an interpolation could be done. For instance, for a = 0.10 the greatest percentile which could be computed using n = 50,000 and the median plotting positions is found at the 45,000th order statistics which is th percentile, and the 45,001th order statistic would give the th percentile. Using an interpolation, a simple linear approximation can be done between these two points. And approximate value corresponding to a = 0.10 or 9 0 th percentile can be computed. Next paragraph will explain how this interpolation could be accomplished. Any straight line can be expressed in the form of S-mz + b (21) 4-5

49 where m is the slope and b is the intercept. Then using the general interpolation formula, Yi+i - Yi Zi+1 - Zi b = Vi - mzi In calculation of the critical values, percentiles or in other words (1 - a) are dependent y variables, and the critical values are the values of the independent z variables. Modifying the general formula (21), the critical values are found by critical value = (1-a)-b The critical values for the standard and the reflected tests were found using this method. For the ease of the computer code, if the consecutive points were identical, one of them was multiplied by and the method was implemented. The reason for this procedure is that in case of identical consecutive X points, computed m would be infinite. Therefore, performing the multiplication the problem could be prevented. For the standard and the modified tests, critical values were computed for the significance levels of a = 0.01,0.05,0.10,0.15,0.20 and for each n = 5(5),50 sizes. However, since to conduct a precise sequential test and find the a levels for it, all critical values at least for a = 0.01 to a = 0.20 are needed. Therefore the codes were modified and the critical values for a = 0.01 to a = 0.99 were computed. In some studies, using extrapolation yo and y,,+i were computed. But, because of the very large number of replications, these order statistics carry the information which is beyond the scope of this thesis. For example the first order statistic gives the percentage point which nobody would need this much detailed information. Since the linear approximation between yj and the yo would give information for smaller percentage points, there is no need to include yo in this study, 4-6

50 neither y10001 because of the same reason. Therefore, for the ease of computer codes they were not computed Reflected Test. The only difference between standard and the reflected tests occurs after the second step. Figure 4.2 shows the flow-chart of the critical value generation for the reflected tests. After the MLEs are estimated using the original sample, in reflected test, each deviate is reflected around A, the MLE of the location parameter. First, the deviate is subtracted from the A and then the difference is added to the I. For example, if the A of the sample is 1.24 and the deviate is 65.83, the difference is Then is added to the A to get the reflected value of For the deviate of with the same ý, the reflected value is After the reflection, the sample size used in computing the CDF and the test statistics gets doubled, but the rest of the procedure remains the same Sequential Test. The method used for the sequential test is different than the other two types. The critical values for the sequential tests are computed in a similar way with the power study described in the next section. Since there hasn't been any computer code published in the literature, it will be useful to explain the procedure for others' judgment and for future use. The sequential test uses two different independent tests sequentially and generates its own significance level. The procedure has been described in Figure 4.3 as a flow-chart. The first five steps of the procedure is the same as the critical value computation for the standard test. The difference starts after the test statistics are obtained. The two different test statistics are compared to the critical values at specific a levels. This procedure is done for each of the 50,000 samples. The number of the samples passing each test at those specific a levels is counted. The ratio of the number of accepted samples to 50,000 gives the percentage point. Then, subtracting that value from one would give the alpha level. To make the understanding easy, let's assume we keep track of the testl at a = 0.05 and test2 at a = If a 4-7

51 50,000 repe Jtos[ Generate n Random Deviates from Cauchy(0,10) [ Order the Variates Calculate the MLEs for Scale and Location Parameters SReflect the Variates Around the Location Paaee SCalculate fth Hypothesized CDF Values --1 Calculate the EDF Statistics Order the Statistics [ Calculate the Critical Values ] CSTOP) Figure 4.2 Flow Chart of Critical Value Generation For Reflected Tests 4-8

52 sample passes both tests at given levels, count is increased by one. This procedure is repeated for each of the 50,000 samples. Then the significance level is calculated as = -# of accepted samples (count) 50,000 To have more precise significance levels for the sequential test out of two different independent tests, the individual tests have to be applied at wide range of a levels. For this reason the critical values for the standard and the reflected KS and V tests were generated at a = 0.01 to a = 0.99 by increment Since one sequential test in this thesis includes reflected CM and V we had to regenerate the critical values for the CM for both the reflected and the standard case. The median plotting position method and 50,000 repetition were applied to this study, too. The computer code for the sequential test is harder than the other tests. Because, for each a level of one test, the other test has to be examined at each a levels from a = 0.01 to a = But, since a levels greater than 0.20 are not frequently used in hypothesis testing, we include only a = 0.01 to a = 0.20 in this research. On the other hand, to reduce the amount of the calculations in the code and therefore the running time, I have developed a matrix-kind data structure to compute a levels. The idea which this method was based on is that if a sample passes the test at a specific a level, then it will, for sure, pass the test at a levels smaller than that particular level. Because, as the a gets smaller, the critical value, however, gets bigger. On the other hand, since the sample passes the test if and only if the test statistic computed is smaller than or equal to the critical value at that a level, it already satisfies to pass the test at smaller a levels. Therefore, 100 times the highest level at which a sample can pass the testi is attained as I of that sample. And J is attained, in the same manner, for the level of the second test. The reason of multiplying by 100 is just to get integer numbers which will serve as the index of the matrix defined below. After attaining I and 4-9

53 Generate n Random Deviates from Cauchy(0,l0) 50,000 LSOtO0i Order the Variates Calculate the MLEs for Scale and Location Parameters Calculate the Hypothesized CDF Values Calculate both of the EDF Statistics Compare EDF Statistics of Each Test With the Critical Values O.W 0.W If o.w ~test2 at jj. S A =A +1 I i*100j*100 i*100.j* Calculate the Critical Values ot= I- A i*,mi*lnn I 50,000 I Figure 4.3 Flow Chart of Significance Level Generation For Sequential Tests 4-10

54 J to the sample, all (i,j) elements of the matrix are added 1, where i = 1,..., I and j = 1,..., J. This is done for each of the 50,000 samples, and at the end, the (i, J) elements of the matrix will give the total accepted numbers of the samples at a = i level of the testi and a = i level of the test2. After this, it is easy to find the significance level of the sequential test for that particular combination. Then the a level of the sequential test for the combination of testi at a = i and test2 at a : is found as asequential test , 000 where (ij) represents the (i,j)th element of the matrix created as an output. The codes were written so that the rows would represent the a levels of the Kuiper test, and the columns would represent the other test corresponding to the pairs mentioned before. 4.3 Power Study Power of the Standard Tests. After the critical values are determined, one other important concept is to check the power of the test against the alternative distributions. The significance levels give the probability of rejecting the null hypothesis when it is true. One would like the reduce the probability of rejecting H, when in reality it is true. Also, one would like to increase the probability of rejecting the H 0 when in reality H. is true. The latter gives the power of the test. The power indicates how good the test is against specific alternative distributions. Therefore the power of these tests were examined against following alternative distributions: 1. Cauchy distribution C(0, 10) 2. Normal distribution N(0, 10) 3. Exponential distribution 4. Beta distribution B(3, 3) 5. Gamma distribution with shape =

55 6. Weibull distribution with shape = t-family with degrees of freedom of 1, 2, 5, 10, 15, 20. Figure 4.4 shows the detailed flow-chart of the power study which is also a Monte Carlo analysis. Basic steps (1-5) are the same with those of critical value computation. After obtaining the test statistics, they are compared with the critical values corresponding to the a level of interest. The ratio of the total number of the rejected samples to the total number of samples (50,000) at a certain a level gives the power of the test against that alternative distribution at that a level. For the power study, again 50,000 independent samples were used for consistency. And the power analysis has been accomplished for the a levels of 0.01,0.05, 0.10, 0.15,0.20 and sample sizes n = 5(5), 50. The steps are explained below. 1. Step : Random deviate generation. IMSL library has very rich number of random generators. For the alternative distributions used in this thesis, samples were generated using the IMSL subroutines. The alternative Cauchy samples were generated with the RNCHY as before. But different seed was used to have independent samples. Thus, the real power of the test could be checked along with the accuracy of the computer code. Except for the alternative Cauchy, the same seed was used for all the other alternative distributions to get more precise comparison. For the normal deviates, subroutine RNNOR was used and N(0, 1) deviates were generated using inverse CDF method [16:1017]. Then the deviates were added with 10 to get N(O, 10). The RNEXP subroutine was used to generate exponential deviates with the antithetic inverse CDF technique [16:999]. Beta deviates were generated using the subroutine RNBET. The algorithm used by RNBET depends on the values of the parameters p and q. "Except for trivial cases of p = 1 or q = 1, in which the inverse CDF is used, all the methods use acceptance/rejection" [16:993]. p = 3 and q = 3 were picked for the power study against the Beta distribu- 4-12

56 START 50,000 Generate n Random Deviates repe tionsc From Alternative Distribution Order the Variates ] Calculate the MLEs for Scale and Location Parameters] Calculate the Hypothesized CDF Values Calculate the EDF Statistics Compare Test Statistics With Critical Values Calculate Power Against Alternative Distribution STOP Figure 4.4 Flow Chart of Power Study For Standard Tests 4-13

57 tion. The Gamma deviates were generated using RNGAM which uses different algorithms. For instance, for shape parameter of 0.5 the squared and halved normal deviates, for shape = 1.0 exponential deviates are used [16:1003]. For this study shape was picked as 2. For the Weibull deviates, RNWIB was used with antithetic inverse CDF technique [16:1025]. IMSL doesn't have any subroutine to generate t deviates. However, if Y has Standard Normal distribution (N(0, 1)) and Z has a Chi-squared distribution with v degrees of freedom (X') then X = Y1V /V is t-distributed with v degrees of freedom [5:164]. Since, IMSL has RNNOR and RNCHI which generate the standard normal and Chisquared deviates respectively, this algorithm were applied. First, the normal deviates were generated and then the Chi-squared deviates were generated. Then the ratio of the normal deviates to the square root of Chi-squared deviates over the degrees of freedom was taken as a t-deviate. 2. Step 2-5 : The same methods used in critical value computation was used. 3. Step 6 : The test statistics are compared with the critical values at certain a levels of corresponding sample size. The power is determined by the following equation # of rejected samples power = 50, Power of the Reflected Tests. As in the critical value computation, the only difference between the standard and the reflected tests is that the sample is reflected around the location parameter, A, after the estimation. Then the new doubled size sample is manipulated as explained above. Figure 4.5 shows the procedure as a chart Power of the Sequential Tests. The power analysis of the sequential tests uses exactly the same algorithm as of the significance level computation as explained in Section The only difference is that, instead of generating only the Cauchy variates, we generate the other alternative distributions. The flow chart of 4-14

58 501,000 Generate n Random Deviates repefltons [From Alternative Distribution Order the Variates Calculate the MLEs for Scale and Location Parameters] Reflect the Variates Around the Location Parameter Calculate the Hypothesized CDF Values Calculate the EDF Statistics S Compare Test Statistics With Critical Values I Calculate Power Against Alternative Distribution 3 j CSTOP Figure 4.5 Flow Chart of Power Study For Reflected Tests 4-15

59 the power study for the sequential tests is shown on Figure 4.6. The power studies for the sequential tests have been accomplished against all the alternatives mentioned in Section but the t-family. Again the samples were examined at each a level of the both of the tests and results were derived again in the same matrix form. Then, the ratio of the total number of accepted samples to 50,000 were subtracted from one to obtain the power of the sequential test at that a level corresponding to that combination. The conclusions about the tests will be derived depending on the results of the power studies and will be presented in the next two chapters. 4-16

60 tstart Generate n Random Deviates 50,000 From Alternative Distribution repetitic" Order the Variates 4 alculate the MLEs for Scale and Location Parameters 4 Calculate the Hypothesized CDF Values SCalculate both of the EDF Statistics I Compare EDF Statistics of Each Test With the Critical Values if o.o _.w ~jsc3ii+oi>.2 P.w 20 test2 at j=j+.o J- A =A +1 F *1 00 j*100 i*100j*100 I Calculate the Power Against A i_1 * Alternative Distribution power= 50, I Figure 4.6 Flow Chart of Power Study For Sequential Tests 4-17

61 V. Results This chapter includes the critical value tables and the power study tables as outlined in the previous chapter. For each test, critical values were generated for the sample sizes n = 5(5)50. Any one who wants to check the data in hand whether it comes from the Cauchy family can easily use the critical values generated as a result of this research. Basic steps of this procedure includes the following : 1. Calculate the MLEs from the data using iterative method. 2. Using these MLEs calculate the hypothesized distribution function. 3. Determine which test you will apply and then calculate the corresponding test statistics using equation (17), (18) or (19). 4. Choose the appropriate table corresponding to the test picked and the size of the sample. 5. Find the critical value corresponding to the a level across the top row. 6. Compare the test statistic with the critical value : "* If it is smaller than the critical value then you fail to reject the hypothesized distribution "* If it is greater than the critical value then reject the hypothesized distribution, with an error level of a. If at step 3, any one of the reflected tests is picked, then the sample has to be reflected around the MLE of the location parameter. After that, the procedure remains the same accept for with the doubled sample size. For the sequential tests, after computing the test statistics, refer to the appropriate table and determine the a level of the test. Then find the corresponding a 5-1

62 levels of the individual tests from across the top row and the far left column. Then apply those individual tests separately as explained above. If the data passes both of the tests at those levels then we accept the hypothesized distribution. If data fails in either one, then we reject the hypothesis that the data comes from the Cauchy family. To determine which test is appropriate for the purpose, power study tables stand as a key. The following sections will present the tables of the critical values and the powers with the necessary information. 5.1 Critical Values The critical values for the standard tests were generated for n = 5(5), 50 and a = 0.01,0.05,0.10,0.15,0.20. These are shown on the Tables For anyone to be able to apply sequential tests critical values for a = 0.01 to a = 0.99 is needed. Therefore the probability points from which the significance levels could be derived by a = 1 - pp are presented for those tests used in sequential tests. Probability points of the KS and V tests are presented in Appendix C. One discussion and disagreement on the critical values could be that the procedure could be affected by the choice of the plotting positions and the choice of seeds. Although this is partially true, having 50,000 iterations reduces the effect of different plotting position methods. On the other hand different seeds don't change the derived values significantly. To demonstrate the difference which could occur by the different choice of seed or plotting position methods, the codes were rerun with these modifications. The arbitrarily picked results shown in Table 5.3 indicated that the first three digits are significant. For those who would believe this was just a coincidence, the variance and the mean of the test statistics were computed for the standard tests using the IMSL subroutine UVSTA. Since each repetition produces an independent variable, the 5-2

63 Critical Value Tables For KS Test Sample Size II Table 5.1 Critical Values of Standard Kolmogorov-Simirnov Test I Critical Value Tables For Kuiper Test Sample Size Table 5.2 Critical values of Standard Kuiper Test 5-3

64 mean of these variables is normally distributed and has the variance of -2 where o, 2 is the variance of the each variable. UVSTA computes a, from the 50,000 independent values. The subroutine uses n - 1 in the denominator [16:26]. But for the sample size of 50,000 it needs to be modified so that it uses n in the denominator. The modified results were used in computing the confidence intervals. The confidence interval were picked as 0.95 using 2o- around the mean and are shown in Table 5.4. These confidence intervals support the experimental results explained above. That is, the first three digits are significant. Plotting seed n = 5 n = 201 n = 30 n = 40 n = 50 positions a 0.01 a=0.15 a=0.20 a-=0.10 a-=0.01 KS critical values Median seed Median seed Mean seedi Mean seed V critical values Median seed Median seed Mean seed Mean seed Table 5.3 Comparison of different seed and plotting positions Examining the tables reveals that the critical values at each level decrease as the sample size increases. But the decrement reduces as the sample size increases. This shows that if the sample size is increased to 70 or 80 there is a high possibility that the critical values would become stable at certain values. In other words, it reaches the asymptotic values. The critical values for the reflected tests are shown in Table 5.5 and Table 5.6. Those tables show the same kind of behavior as the standard tests. One significant result of the critical values for reflected case is that the critical values of the Kuiper test are exactly twice of the KS tests' critical values. The reason of this can be explained analytically. Since reflection method makes the 5-4

65 Confidence Intervals (1s2uT) Standard V Standard KS n Upper Level Lower Level Upper Level Lower Leve Table % Confidence intervals for the standard test critical values sample exactly symmetric around the location parameter, each original data has its shade on the other tail of the sample. Therefore, the difference between EDF and CDF is the same for the distance below CDF (D-) and above the CDF (D+). This causes V which is (D- + D+) to be twice of KS which is max(d-, D+). Then the critical values come up to be twice of the KS test's. Sequential tests were generated for the same sample sizes as with the other tests. But the individual levels were applied at a = 0.01 to a = The resulting significance levels were displayed on Tables for CM - V, CM(Ref) - V and KS - V respectively. For any significance level of the sequential tests, the critical values are found from the corresponding tables of the individual tests at the corresponding a levels which makes that combination. The critical values for the CM and CM(Ref) were regenerated for a = 0.01 to a = These values are presented in Appendix C. 5-5

66 Critical Value Tables For Reflected Kolmogorov-Simirnof Test Sample Size Table 5.5 Critical Values of Reflected Kolmogorov-Simirnof Test I Critical Value Tables For Reflected Kuiper Test Sample Size Table 5.6 Critical Values of Reflected Kuiper Test 5-6

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82 5.2 Power Analysis The power analysis helps to examine which test is more powerful and efficient against specific alternative distributions. There are three common results of the power tests. First, as the a level increases the power increases. This is not surprising, because as the type I error increases, type II error decreases, therefore the power increases. The second common result appears to be that as the sample size increases the power increases. This is also a common sense, because larger samples carry much more information than the smaller ones. Finally, the theory that the Cauchy is a member of t-family with degrees of freedom 1 was ;.roved by the power results of the standard tests against t(1). The powers are very close to the corresponding a levels even for t(2). On the other hand, the idea that the Cauchy would give a good approximation to the normal was partially proved, too. But, as the sample size increases more than 15, it becomes obvious that the power goes up. That is because the larger samples help to distinguish the larger tails of the Cauchy distribution. So, it could be, with a high confidence, said that the Cauchy could be used to approximate the normal distribution with a sample size up to 15. Next sections will discuss the powers of the three test types Power Analysis of the Standard Tests. The power tables of KS test were presented in the Tables The results for the KS test support thý- conclusions of Ocasio's thesis. KS test has very high powers against both the symmetric and the non-symmetric distributions. Compared to the CM and A 2 tests for.e standard and even sometimes for the reflected cases of those, standard KS is more powerful. The results for the Kuiper test presented in the Tables show that V is the most powerful test among those studied so far for the Cauchy distribution as hypothesized. V has at least twice the power of KS against the t - family. 5-22

83 Specifically, while the power is around twice of the KS at a = 0.20 and a = 0.15, it goes up a lot more than twice for a = 0.01, a = 0.05, a = For example for t(5) and n = 50, while the power of KS is at a = 0.01, the Kuiper has which is almost 18 times better than the one K-S has. This ratio goes up to 20 for the t(15) and t(20) at the same level and for the same sample size. For smaller samples like n = 10, 15, V has almost 5-7 times better powers than KS does. The same behavior of the Kuiper test is observed against the Normal, Beta, Gamma and Weibull distributions. For the Gamma, even though the power doesn't go up as much as in the other distributions, it is still around 150%-300% better than KS. Since the Gamma used in this study is non-symmetric (but not too skewed), KS has better power than it has against others. For exponential distribution, both tests have approximately the same powers with KS having slightly bigger values at larger a levels and V having slightly better values at smaller a levels (a = 0.01,0.05,0.10). It has to be mentioned that both of the tests reach their highest power levels against the exponential distribution. In general, after sample size gets more than 25 for every a level, the powers fall in the range of The power results against the exponential and the Gamma show that both tests are very good against non-symmetric distributions like exponential. But, V test has better power against non-symmetric but skewed two tail distributions Power Analysis of the Reflected Tests. The power results of the reflected tests are interesting. The powers of both the KS and the V tests turned out exactly the same for the reflected study as seen in the Tables But as explained above V statistic has always twice of the value of KS statistic, and the critical values for V are twice of KS'. Therefore, even though the values are different, because of the same ratio in the statistics and the critical values, the powers turn out to be the same. 5-23

84 On the other hand, the expected result was reached with the improvement in the power compared to the standard cases. For symmetric or nearly symmetric distributions (in this study t-family, Normal, Beta and Weibull), the reflection technique increases the power. The reflected test method doesn't get any improvement for the sample size n = 5. In fact, it resulted worse for the V test than its standard test. But as the sample gets bigger, the improvement in the power starts showing up. As noticed from the tables the improvement in the reflected V test is not as much as in the reflected KS test. Because, even for the standard case V test alone has real high powers compared to the KS test. Even though they both have the same powers in the reflected case, the KS has much more improvement than the Kuiper because of its relatively lower power in the standard test. The reflected test doesn't improve the power against non-symmetric distributions. Examining the powers for the exponential distribution in reflected tests reveals that the power goes at least half way down compared to those in the standard tests. This result was expected prior to the study. Because the intuitive analysis would indicate that even if the sample is not symmetric, reflecting it about the location parameter would make it perfectly symmetric anyway. The same kind of reduction in the power is observed for the Gamma, too. Because the Gamma distribution picked for the power study had shape parameter 2 which makes it non-symmetric. But the reduction is not as much as in the exponential distribution, because although the Gamma is not symmetric it is still two tailed distribution. However the power is still low compared to the standard tests Power Analysis of Sequential Tests. The analysis of the sequential tests is much harder than the other two types. One reason for this is that the sequential tests doesn't have exact significance levels such as a = 0.05 or a = 0.1C Very close levels were derived, however each level closer to those exact a levels have different combinations of the two tests. And different combinations resulted in different power levels. For example, very close levels to a = 0.10 give different power 5-24

85 levels in the range of for the sequential test of CM(Ref) - V against exponential. But the limits of this range are determined by the extreme points as seen on the graphs. The graphs show that the real range after disregarding those extreme points is around The closer examination shows that the variance in the power differs from test to test and depending on the alternative distributions. The CM - V sequential test gives very small variance in the power against exponential and relatively small variance against the Gamma distributions. It seems like since both tests have higher powers against non-symmetric distributions the sequential test turns out to be more powerful against non-symmetric distributions. On the other hand, even though the Kuiper test is powerful against symmetric distributions, CM's less power causes the large variance in the power against symmetric distributions. In the combinations, as the a level of V decreases and a level of CM increases, the power goes down. The power study results for this test are included in Appendix D. Here only the results for n = 25 and n = 50 are presented in Tables along with the graphs (Figure 5.1 through Figure 5.4). Moore and Yen showed that the reflection technique improved the power of CM test against symmetric distrif.utions [23). Therefore the sequential test of CM(Ref) - V turned out to have very low variance in the power against symmetric distributions. The complete power results of this sequential test are included in Appendix E along with the graphs. Since the reflection has negative effect on the power against non-symmetric distributions, the large variance in the power for the exponential and the Gamma is observed in this sequential test. But for all of the symmetric distributions the power has very low variance. The power results for n = 25 and n = 50 are shown in the Tables The powers are plotted for these cases in Figures The last sequential test which is the combination of KS and V has the same kind of behavior as CM and V sequential test. Because of the relatively low power 5-25

86 of KS against symmetric distributions, the power against those has large variance depending on the a level combinations. But both tests have very high powers against non-symmetric distributions. Therefore powers against the exponential and the Gamma turned out to have almost no variance. The results for n = 25 and n = 50 are shown in Tables and corresponding graphs are presented in the Figures The complete results and the graphs are in the Appendix F. In the graphs, z-axis is the significance levels and y-axis is the power levels. The continuous lines represent the power level of the sequential tests. "o" represents the power levels of the Kuiper test while "'" represent the power levels of the other individual test used in the sequential test. For symmetric distributions, the power of a sequential test at any a level is somewhere between the power of the two individual tests at the same a levels. This indicates that sequential test improves the power of the individual test other than V, while reduces the power of V. On the other hand, for non-symmetric distributions, sequential test reduces the power for both of the tests. As seen on the graphs, the power levels of each of the three sequential tests are lower than the power levels of the individual tests against the exponential and the Gamma distributions. 5-26

87 [In Power Study Results For K-S Test a Dt Cauchy [ Normal Exp J Beta Gamma Weibull Table 5.10 Power tables of Standard Kolmogorov-Simirnov Test against alternatives 5-27

88 Power Study Results For K-S Test " aiiii Cauchy [ Normal Exp Beta Gamma WeibuU Table 5.10 (Continued) 5-28

89 IPower Study Results For K-S TestI n ca t(1) t(2) ( t(5) t(10) t(15) t(20) j Table 5.11 Power tables of Standard Kolmogorov-Simirnov Test against t-family 5-29

90 Power Study Results For K-S Test n II a t(l) rt(-2) t(5) t(10) t(15) t(ý20) Table 5.11 (Continued) 5-30

91 IPower Study Results For Kuiper Test n a Cauchy I Normal Exp Beta Gamma Weibull Table 5.12 Power tables of Standard Kuiper Test against alternatives 5-31

92 Power Study Results For Kuiper Test n II Cauchy Normal Exp Beta Gamma WeibuU , d l Table 5.12 (Continued) 5-32

93 Power Study Results For Kuiper Test "n a t(1) t(2) t(5) t(10) t(15) t(20) Table 5.13 Power tables of Standard Kuiper Test against t-family 5-33

94 Power Study Results For Kuiper Test n 1a 11t(1) t(2) t(5) t(10) [t(15) t( Table 5.13 (Continued) 5-34

95 Power Study Tables For Reflected K-S and Kuiper Test n 1J a D Cauchy Normal Exp Beta Gamma WeibulU Table 5.14 Power tables Reflected KS and V against alternatives 5-35

96 in Power Study Tables For Reflected K-S and Kuiper Test ci a] Cauchy Normal Exp Beta Gamma Weibull , Table 5.14 (Continued) 5-36

97 Power Study Tables For Reflected K-S and Kuiper Test] - II 1 o 1 t(1) I t(2) I t(5) [ t(10) I t(15) I t(20) Table 5.15 Power tables Reflected KS and V against t-family 5-37

98 Power Study Tables For Reflected K-S and Kuiper Test n a t(i) t(2) t(5) t(10) t(15) t(20) Table 5.15 (Continued) 5-38

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100 NO. - o- O@~ ~ *1#.U ~ ~~~1 P1 P Ii 1fW * ~ ~ D ~ 111 ~ 1 0 on0~~, ea W m.00. ~~ I* 6 U U 'W 33. *a*.e fo 1100 S0 1 ) IS 00 W.010&0,00 0 0e * * q.~ci Dq V 'a 'a 41 W-q.000 R..0!m W v01 m1w1-wv 0 10W a 0.01pit001I w 0o... D o0 10 r 0W.e, nd 0104WO C.O O11 *000a000ae** i ~ ~ 0 ~ ~ ~ c.001i0l.0..~q*.* t- a W "IT I *!WO C. 01 *W~ ~~~~ 10~ W W 1 M 0~ Ch o O 1 W '.- v ta 10 g0 0110a 00 "C4 a- -.. ec,1~ W.. 1. ; " ' 0 w,0 =-CIW.@.. %;Z-;!Zo1#00 0 &9 gg000. aof11"d1. 11 e 0A 090 C-4lev a6.0 v n 00 n-00 0~~~~~~ *1 0 1 ~~~~~ 1 1W W 010 ID.@ W00. l.0 DO D = = =4100W -.o FI I0 ft0i0f00000 W00 I OW 10 ~c ::=b90%00 ~~~~~I 01W10 01 a * a0.e.o 0 cbocn, w *0000W01 W 0004P : 11" -~ ~~~ O O ! 0 s W 0 1.I 0 W I o; 0 0. c%. so O.:0 ww w- o * Po a ooeoo Z

101 Power of CM-V against NORMAL for n= o ! ' p..v.... S *: pw of CM: v o... I 9...l...! Significace Levels.... Powerof CM-V against NORMAL for n I i..." :... I... :pw of CM *... pe of Seq CM-V Significaace Levels Figure 5.1 Power comparisons of CM - V against Normal 5-41

102 C. 0. a000d!a *** 0~~ *01"~O~ ~ Pt r ~~l0 r. 0 t ~ ~. ~~~IV rt0: s.c R ~. e.. eta 4!~o~o e! *-s,-ý 000 4!0 0.0~ b-o 00 O a p00 I,- P* r*. * *~ *~ 671--P *ooooa.0 So r- o wa oo a O I3gr-M I - j -. ~G 0!b C~et00., i C G! 4 0! C1 a,00 c PII e~.0 e 0.. tt e.1 500Mae U. 0Q asm * e 00 O O t ~.. t....-!c! C t4 t4!4 1 M ar A. A08v00000 an 04. o G! C! 0! 4!ooo ooa! 4! G!qo 0t 0qo o. A Q!4!. C!4! 0 10 a 0 11o oos oo. -W f~b CC. 00 Ct a0 a, a a a 4. a o a a o 14! 0! e m r I; I'%I q v,00 -- I In, Z &TIE a 9S Z10, * I-: 1.C.0 a' '0 n 0-7 r 5-42 G"!.....

103 Powe of CM-V agant EXPONENTIAL for n= : :...:... S Sb: pw of CM ; : Significce Levels Powae of CM-V aains EXPONNIAL for no : ' :..... : : o pw ofvm 0.4'. Lpweof~ CM- 0 ~o of V 0 I i I l I Significace Levels Figure 5.2 Power comparisons of CM - V against Exponential 5-43

104 ~ ~~~P p P... P 00.0 * ;p P rsa. *1o o o oo o 0 o -.0.ee4.~~ e. 0eq ~ 0.0q a0q.-..e.0.o000..e o o o~~ 00.t q q ~a. 1-0C n 01 MOM o M 0... eeqw.00.0ee.a V.11 a-.1.. C; C. r. I ; Ii ;4 :: -x P r c;.oe0a0..e.1".i-e.0 M IM q n.~o do. - O -001"0 4a O,.0.0q.aqq0o0.0 'a a ma 11. 'll.0.0 e0vqe n I, VV.00-e.00a.OeG!.0 a-ae J.0 a ~~~~~~~~R q0 00 e~00~q -oqa.n.4.0eq0 :43a bqe *0f 0' oeqeqeqep.ni.00qep.03 04b 00 on VO 4 ~ O~.eqnq!Olit..0(l0W0Ia.G! 5-44D -4 0.

105 Power ofcmyvagainst BETA for n;=25 00 * PwofCM: pe of Seq CM-V Significance Levels Powet of CM-V against BETA for R Figure Powe copaisn ofc VagistBt

106 O* a 13! 3! - *. 2 'a 2 Oa.10* o t Mae~ *n~a.on. v ac v... ~ i.-0 * b t t~ -i:. q 0 a az ~1 0121:*0, ; -.0 x )Ott t Wl'!1. 11 a..-...d... 0 e~ 0 0U;Z g! *0fl044a-r C. 'a.* 0 **a e. c..-.n ~ ~ i Re R t I t1 1 1! o Voa na a a Id!. id!q!4 00! 4 -~~ i-a-a- CIS ~.4 (i a- 0 : a w i. o O.em 0.~ ~~ ~~~~..0o o SO0 0 c; C ~~~~~~~~~~. 30 rz a00 0" 00404:4 s0 00NI0) : ;. e q ~ a. o4 0 a 4 C O Oe O 0-000,4001-.V" C10 X0 0 Via000q v *0 A0 V 04 IS0 'm , 0 *p-l )- 0 V.40 *) av-- ef 4 C ~ U U C! G! 0.~ C!4U 0:029:9:"0 5-46

107 Pbwerof CM-V awinst GAMMA for n=25 R 8....i i Q S " *' **... 0 pw of CM: :... "... *... pe of Seq CM-V Significace Levels Powerof CM-V against GAMMA fr n--0 I *,...,,,,, !.' Significance Levels Figure 5.4 Power comparisons of CM - V against Gamma 5-47

108 I a A 0! -S i -~ol.::.. v ;C;-.02 t- s.* *.. qa.@l. ao I* t" v 23 LQ, on a 4 v av c; 0 I Soo -to-.2. 2*A 9- o ; 0. q. c lo 23'02 So * 2 X 0 *.ml aa - aa aaac.~ a * ~ -. ~ e aaa-. ~ ~ q ~ ~ aq ~ aa.. U,~~~ ~ (.a~u X'' e lo 00U0 o au 0 aa c.4.,~ ". a Sx au, ".'a Zoa o ~ ~ a U ~ U~a q ~ ~ o. U'a 0 'ince.1 asaa a 00.aUx I vwd I n o oo m 91; z w 5-48a

109 Power oi-v M again WEIBULL for n [...:..... i : i... 0o.6... i... oi.... i... * ' : S: * pw of CM: 0.2 ~ ~. ~ ~.. ~ ~ ~... ~ : pceof Seq CM-V Significance Leves Power of CM-V against WEIBULL for n= ' !....p.v i... *:pwofcm: Significance Leves Figure 5.5 Power comparisons of CM - V against WeibuUl 5-49

110 2n n11 C4q00111 MA Ii--O A A * i 60 PIPF u.e 0101r10 n 01 el3-2 P Ci."! li I: C! Cc1 C!001!010 *.. ~ ~ ~ C " to 0~~~~~~~~~~~~ 0 a100 a n " ( IV r Ile t001 in c R On1,0 ~ 0o n* C, Cc a I1 0 I0 I 01i I 0 00!0. 0 0!C 0 0. C4 C! "Co CD ip 1010 A vv O ý nfv A, IV, P, P I` P V S z010"a ,010 % 0 N In I a00 A a :0 30 e a 00C, 1 F c, I A a'p e e VO ` 1 - c5ot ".,.-;!- r,

111 i~. *~ ~ ~~~~ w..~04 1Ad 0-!w-04 4 s r!6 A e eq 4~ A N~e f on 100 e. 4.. c q 4!3 n4!. 4 Pi-*- ~.n ~.a-.# 6. 0 a 0 a-n.-c eo04s 4444 C,n 4.E,.fl OC ln lflo a.o 4C) Oi; 9 ' ;x 404 W0~l : v.~~ c ir 4!4 4!..Ca a..*....4.@.4. o~p e~w4... eee. 44~CQ@4C44aeC* Pae8..C Ie~ol * 4~W *444.0flC~flC@OfflP P PC I--;--.0 I~i0 )flccwwcccccccc@cc~ 4 C.I** O e C r ~ 44 toml oa0s " a ob :a t-**e ee** wf4 "Ik.Of0Cs 'IV OS 8.C a-@0444q0 11 Li eo~~ 00 OrOO *.C0 OO OOOl~ 5-51,I*Ic

112 Pow of CM(ReO-V againt NORMAL f n=25 III I I I 0.O i D: ;... PWofCMet 0.4 pe of Sq CM(ref) Significance Levels Power of CM(R f)-v againt NORMAL for n=5,, i :.... i '. i:...:...!-o... o) 0W i.... :wofvc~fl v i :peof SqCM(reft) Sigaificance Levels Figure 5.6 Power comparisons of CM(Ref) - V against Normal 5-52

113 a *1.. a *** ** *.00 ~ * e to 0 2 we... r -. I;q. f. ae.e ~ "2141nee 00k a. *a. OC W 6:.... WfO 0 a. a a.. a b. -~ 000I3.0~o0, 0. a q~ * 0. g.10 0 * ta0 C o00 40=C~ 0 0l*d0E aa ,0 r; P a q 0.*0 * 0 f. 0. ~ ~~ i! C V q Ar * 0 O f0 :Q! 08-e88C.0..0.a C *. d! 4! 4! a 8-X ~ a"ee ;9 M vv00088 "Cone~a a. e beq. *..e Z0 "R a,.. * O NS8e:..OIP No. 408C.0 CSa0 C I.00 SS 0. A4.)VV.co- N M O. U. 0.C o-e ~ eu....0.s...0w C v k a I - - e m.0 1, F PS Im~. I.; 0 I C! )C f8 C.0 C ID.(9. 0 fl 0 9C! d d!q!! C d!4!! o **.C04@C.. e N. C. ~ C n~i0 Ut - - *g! LJ 0 OP"T :101 e F,.00 6 *S....0 V.8. -P C. 0S.*.f a0 nso"' al a a;. a a a a 0. e o o o.oo o.. o e.. e.. e.. 4. we ;; 5-53r,9 3;i

114 . Powe of CM(Rcf)V aid EXPONENTIAL fo n= !' :.. S... ~ i... i Q b ' !......!.... A i.,..:...,... :... i...! A :pw ofcm(ret) L: pe of Seq CM(re-V Significace LveIs 1 Power of CM(Rcf)-V 2gahb EXPONENTIAL for n " p : : :*"pw of Ckireo :... *_: pe ofa S (ref)-v SignificanceLevels Figure 5.7 Power comparisons of CM(Ref) - V against Exponential 5-54

115 I 140G 0 I-. If 1!4! 14 laas 0 a a a d!4 -. C000CUW*f -. -~0 P r o Uý riupppr; ;z 1! I. Is6 40r; -PP.*Pa ri I* * * * * * P 0P 0 a p 4 a a~~~~~~ e X-.ao N~o.00 a g e nas wo. e no.v ~.. 0 0a CD000 w 0 G fo~ -1,o w a-e~q Is! a!a4!-!. d!4... oa-e~~~~~~~~e. UGWC OS O IF C G O ~ - e o C o 000~~~ls e w ao. O a ae~ agac 0! 04! CG0GOGG~~~~e.Owno:, e C Gf w 0..o1 of0. 00 = :awn-w...o Oo. 12r G..OOG 1-GO a.~~oogog~ owo a.oo.0000w.0 a~ a-o

116 Powe of CM(Rd)V apinm BETA fr n=25 1 I I I II uo.6 i S :. : o fb: f ) I!... i... i... i... ;... ::Pw of C1 (M)- *....peofseqcm(ref-v Significue Levels 0 Powe of CMQd)-V against BETA for n=50 0., *.:pw of CM(rt) :peof Sq CM(rcO-V O SignificMce Levels Figure 5.8 Power comparisons of CM(Ref) - V against Beta 5-56

117 MIX 0" WE KV,.... PR I x p P X P, am "S p la p 0-0 too- 0 S -1 "P C;.0 a W. '0 : I - " X "a o?a a X a. S.Ob 3 a 1 t- V Fý 00 W? It I It 4: at at 0- a a a4! 0 a Xv :Ja G! 10! 4! d! 4!... 9 " X8 X -or.1% X *a z Iola 1.0 ; -2 " 014 0" a *1610wr.4 -w P a - 234og a on me it t.. a 01M wl=lg tot. I I ::o:oo 'a xr.-q 11 " P-13 -P 21 Rr 1*7 111 En ss a.3 lo! o V a a o V21.11 a- f "'vo 2a..2 a 1% c.1 11 c o I Ot 11 a a.1 3 a 'i X. r-- r--- f I P Fall a o n Ul: 14%. a a a :a lo am Pf qr -Pr.1; 1- P P P. a- v.6 Sl a -4 c; v as ; i a c. 0. r. c IS low! F " 9, I I It..1 a R T; a -r- P. c... M X 9 z F 2 ;r C a N'a V a P n a act a a e. " 11 nvswo a.6 als C *a-a 2 'a a a. a. a a. a. a 11 a o 11 I r rl 'o c UX C 2 : a a a "a q NO lo no Il I P a ra 2 c. 0 ci v a o:o a... V 'A -6.6 Iola Av 2 I Me, 'S :2 le i c; *- PC 111; 4 a 11 o lc* UR ""0 11 P rq 1; P P P--a 4 r-7-1--" 11 Ira "*I" 4 20 Ica, "ov o 0" f -. 0 R A 'Do 6 c; 21 E., S*lql!iRIR.' WO :'- F.12.am Fax:. WO V ft tt. t c; I lam M a a a a a M! 4- v = 93= 29= 94 3 ý ; r ;1;. v 2 IP N a 9 "m.6 a a , "S 28 *61: 'a 9 G! 4! 4! a! c! X 'or 4! op a! I r 4! "T! 4! R lool"oplipp I.- p I R oc n o'sr= "":P9 owaaa... e. o. f.-. a a. c. WO w2oao;::ooo. :oomo '61 z 11. tigir 11. IR 1; rl X me "o ft c:c.4 'vw -D Is a. t. s a. oe c a " w a.. 2 a Cam -v. o.. c" P P., 1: 11 P I. rl! 'Fl I, loo 'tnsigr ns: V 11 a 11 1, 044 W ID N. a.. e. a n a.6 oooz:2. Q! 4! 4! a! 4! a! w111; ;;P X o on:, A IS DA 2 c aft a J. lp a R a. I o a glify1 a "'P "1*1co a 6. 1ý- g 4 a w!w qlf.4!.q -Xncooo awoo,= olv. = 32"2vo "coca al-1121ffilttlq M e I - el - me a a 1. a t- 4 a *1-3 abooe:ao am moo. 9 A P P Cl 4 n n t Ai 2 X* S 3 S a t 4 A c; CcC 5-57

118 Powe o CM(R -V aghinst GAMMA for n= o pw of CM(rci : ' :pe of Seq CM(*V A Significa, Levels Power of CM(Ref)-V agaimt GAMMA for n=50 0 : , "... ~ ~~...!... : :pwov *.:pw of CM(rCf) : pe of Seq CM(ret)-V 0i I i Signifiucc Levels Figure 5.9 Power comparisons of CM(Ref) - V against Gamma 5-58

119 *0 **VV. 0 a.a.. be* ~~a- *2:~~aa.. a aa~n~naai.va -. a nan a ~a.i 'o* a0aa 0 O - ae-naaa...anab.. 4aeP... 'Piz arapip.. ~ b.n a -I a a a ( q-aw eaa~aaa I'- aiea I -30 b b-b 0000aa~aa to,- U.. X e n a aa n an - c... a a a.a a 6 a a b a-! 1. a! a!a a.... a,4 a 22 0n en 40c a FO.~ 0aa a a o1 a- aa0.~.00 : M.naa Fraaa aabb F ~ -- aaa.a O01b-a~~. aa.2a-.acab.q..ana -a% e;n aaa O~~~~ a ala Can~aaa 01 n...aaa.....m a ~~~ ena ~ d...anaa-~ a 0o am. ~ a,.0ea a a a... :I I c 94e U3.R In

120 0.8 Power of CM(Ref)-V apim WEIBULL fr 7-25 /...i....i * 0 IW ID Of V.-* '.,.. i... U" -... i i...".... *..pw of CMref). PC of CM(ret).v 0 i i i i i i Significance Leve Power of CM(Re)..V Wa I l/ *, WEIBULL for n , *....:peof ecm(ref) Significance Levels Figure 5.10 Power comparisons of CM(Ref) - V against Weibull 5-60

121 C? ft2 4 U t- f ~ U 6 a6 0.fto t a.-. * a 1a.0 *rz 1-2A F600. r P4 00 V~~~o~ t@ e Un 6' n*.6f06 64tn W.0 t ~* 0. l....ft 4 an00 f a W..0 0.t a O 4404 i 46 W60644 W6-0f46 Nt oe.-- - Oe~ *20 no on46s-6.f a W e o on - -- f a* 0 0.~ n

122 ~~~~ a * * C *~a. yyo.c o.0 0 r 1.0fny - 0 0e- a- 0) a-~y e n.o C n.o q. N a ftnnq.1vv.ooe.6.0 n.0o.0. -~..4..e.o 29n o ee. -. y.-. no..... y,. sr. ar-- an.no~ t -yc.ceqn - ~ ~ 0..ecqo~@~.~~ c Cc,.n ~ -e... nn y y aa , ecnoe o~i-. eqen 0.nwe. o.@oonnoww 6a a O. i.u~ 0 o. o.... o...nn...nn. C! ~ o o c 2a~4o c 2.e y.i :o*** *. * n ey...y we, e.ni--oo@ e.. oy.o o ni n w O~ ~ ~ ~ ~~~~~r- 1"'~q~qnoo--a P I".e.y~EOwO no I..I no I eq I I I o. -c.;!eci` r;qww 24.oi00n... e ao.e.nqc-q-.oe O cnn o..e on...a qn. o..n cnt- nnh- ooo n on oc coo o c.n. cn o ~ n~. no..am,-'... n o. o..nyc;c o-nn c -.0e n o a- 0 eq oo o e o....am. G... 4!.0!..m ~ooeoooeo~oo oooo 5-62

123 Power oks-v gainst NORMAL for n=25 as "... :.! : '~ " ,: Pe of Seq ks-v Significance Levels 0.8 PowerofKS-Vagaigsi NORMAL fera= '.4- -." : i., v :pwofKS pe of Seq KS-V Significace Levels Figure 5.11 Power comparisons of KS - V against Normal 5-63

124 !. ~..!......T -11 P "'P ~ -l!9~ P I ANa 0 00 P00* 00 **00* X ~a 00 POO0*~O0**00 It M.na rao...a0 00 a as ~ ~~~

125 Powe of KS-V apins EXPONENTIAL fo n=25 I o I I :... :peofseqks-v 02 0 C I I I I Significace Levels Powa of KS-V agait XOONTIAL for a= I D *.. *pw of KS Ie PC of ki Significace Levels Figure 5.12 Power comparisons of KS - V against Exponential 5-65

126 P PR P6 W....** * Q!... d ~! **.o~ e - a. e!. 0 - P -- rip 1r -1P-r I 3r Iq..waeaeae~qqo.Ffl.. s - s o1 A oi.0e.0 0@ * 0.o. wq C;00 cia 00a on0. I.. I ~~~~~~~~ a nn o ow or~ -en 0im.aow o.....eq. i.. Vq 0"2 1 IS

127 Power o/ks-v againt BERA fn= I i :... Signifi~cce Levels Pow& o KS-V ags BETA forfn o S i:pwofks e:pwofseqks.m Significace Levels Figure 5.13 Power comparisons of KS - V against Beta 5-67

128 01 e~ n *e 4! 4! 4! 4! e..2a.a.a r! In a. - u * * -0 an.. a. ~ ~~~~~44. a.. 4!~-.a.....a.. I O w - -- V - o a k-.aa aa~ a P ~.U s a *a. a-a. 3a R : I I j C, * a I's-. 9. la low : ~c, I :..1 :: ~OOO@OC@12OOO0 0 e aaa o 5-6P8 r

129 Povw of KS-V again GAMMA for n=25 o..r i...!... i ' SignificceLevds Po r oiks-v Yagan GAMMA or n= l; 0.6. f '' i... " I i " !: i: p... o... I i o l... i ;....:...i Significwce Levels Figure 5.14 Power comparisons of KS - V against Gamma 5-69

130 A a 0*106W.0@@l FI; a..~ 4 A 0100 *0 U ** O G S * S x10 *O a In 1,he~~ W.00.e~.q.n In01 0~~~: * a.a, R- F00aj aa v ace a qnv*: I-1-a011 o. 01 e. 0 u a ~ a...a aa a a P F- '00 0 ~ ~ ~ ~ ~ 0. ~ ~ ~ 1 f~a1a~k ao.faa..#aa~0 a.~e ~~~~~~~~~ ci a a n a.a...~et a n..i ' (CMI a * 1 a 11a0 N10S-01v a. a ZZ a eq ~ ~ ~. a~ *1 0oa.. o... n~ gon a ; o~ean 0 C40em.a a Z a a A n~a Plzr~ 0b--0 ft ~eoca.0 Z* a a~.a a...-1 a0. 0 n...#... -!; I- i ki-p a a. N0R0 b 0 C. f4!0- a WO w v C"3a RO 0 Cl :.5-701co

131 Powei dks-v aghast WEIBULL for a=25 0o.6... :... o...! a p... w pweofvks Significce Levels 1 Powe ofks-v agaizd WEIBULL for n S0.4 ~ ~... ~~~...i..... p q :pwofks *_pcofseq KS-V Significmce Levels Figure 5.15 Power comparisons of KS - V against Weibull 5-71

132 VI. Conclusion and Recommendations 6.1 Conclusions The results and analysis of this thesis were presented in the previous chapter for each case studied. The conclusions derived based on the results can be summarized as follows : 1. The first three digits of the critical values for each test are significant with 95% confidence. Therefore the tests are applicable to any samples with the size of 5,(5) KS test has higher power compared to the CM and A 2 tests as Ocasio concluded. 3. V has the highest power against all distributions and therefore overwhelms the other tests. 4. As the sample size increases the power at smaller a levels increase. 5. Reflection technique improves the power against symmetric or nearly symmetric distributions for all the tests. The improvement starts for the samples with n> Reflection technique reduces the power significantly against the non-symmetric distributions. 7. Significance levels of the sequential tests are less than the sum of the significance levels of the individual tests. That is, for the sequential test of c which is the combination of test a and test b, the significance level is act a< + ab 6-1

133 8. The power of the sequential tests against symmetric distributions at a certain a level is some value between the powers of the two individual tests at that a level. 9. The power of the sequential tests against non-symmetric distributions at a certain a level is less than the powers of the two individual tests at that a level. 10. Among these three sequential tests KS and V sequential test gives higher and the smoothest power against non-symmetric distributions. This study offers a close look at the sequential tests. The behavior of the sequential tests follows interesting patterns. The results of this study can help those who want to learn more about the sequential goodness-of-fit tests. 6.2 Further Research Some further research interests as a conclusion of this study are listed below. 1. The study brought out the most powerful test statistic (V) proposed so far. It has been seen that even the reflection method cannot give too much improvement to its power. Therefore, we believe that a modification to this statistic can be brought by way of an improvement in the estimation method. The computer code CMLE can be modified at least by reducing the tolerance and increasing the iteration number. 2. V statistic can be applied to the other interesting distributions such as the Weibull and the Lambda. 3. The relation between the critical values and the sample size along with the significance levels can be investigated. Thus, more general tables can be generated. 4. More detailed studies can be accomplished on the sequential test by increasing the a level range of the individual tests. 6-2

134 5. The functional relation of the sequential test can be computed. 6. The relations of the different tests and combinations involved in the sequential tests can be investigated via 3-D graphics. 6-3

135 Bibliography 1. Amstadler, Bertram L. Reliability Mathematics. New York: McGraw-Hill Book Company, Andrews, D. F. and others. Robust Estimates of Location. New Jersey: Princeton University Press, Bai, Z. D. and J. C. Fu. "On the Maximum-Likelihood Estimator For the Location Parameter of a Cauchy Distribution," The Canadian Journat of Statistics, 15: (1987). 4. Barnett, V. D. "Evaluation of the Maximum-Likelihood Estimator Where the Likelihood Equation Has Multiple Roots," Biometrica, 53: (1966). 5. Bratley, Paul and others. A Guide to Simulation. New York: Springer-Verlog, Inc., Daniel, Wayne W. "Goodness of Fit : A Selected Bibliography For the Statistician and the Researcher." Public Administration Series : Bibliography, Monticello, ILL: Vance Bibliographies, David, F. N. and N. L. Johnson. "The probability Integral Transformation Ehwn Parameters are Estimated From the Sample," Biometrica, 35: (1948). 8. Devore, Jay L. Probability and Statistics For Engineering And the Sciences. Pacific Grove: Brooks/Cole Publishing Company, Granger, Clive W. J. and Daniel Orr. "Infinite Varience and Research Strategy in Time Seires Analysis," Journal of the American Statistical Association, 67: (June 1972). 10. Gwinn, David Alan. Modified Anderson-Darling, and Cramer-von Mises Goodness-of-Fit Tests For the Normal Cauchy Distribution. MS thesis, AFIT/GOR/ENS/93M-07, School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson AFB OH, March Haas, Gerald, Lee Bain and Charles Antle. "Inferences for the Cauchy Distribution Based on Maximum Likelihood Estimators," Biometrika, 57: (February 1970). 12. Harter, H. Leon. "Another Look at Plotting Positions," Communication Statistics- Theory, Methodology, 13: (1984). 13. Higgins, J.J. and D.M. Tichenor. "Window Estimates of Location and Scale With Applications to the Cauchy Distribution," Applied Mathematics and Computation, 3: (1977). 14. Higgins, J.J. and D.M. Tichenor. "Efficiencies for Window Estimates of the Parameters of the Cauchy Distribution," Applied Mathematics and Computation, 4: (1978). BIB-1

136 15. Howlader, H.A. and G. Weiss. "On Bayesian Estimation of The Cauchy Parameters," Sankhy: The Indian Journal of Statistics, 50: (1988). 16. IMSL Problem-Solving Software Systems. IMSL STAT/LIBRARY : FOR- TRAN Subroutines For Statistical Analysis. User's Manual Johnson, Norman L. and Samuel Kotz. Continuous Univariate Distributions-1. Boston: Houghton Mifflin Company, Kahya, Goksel. New Modified Anderson-Darling and Cramer-von Mises Goodness-of-Fit Tests For a Normal Distribution With Specified Parameters. MS thesis, AFIT/GOR/ENC/91M-3, School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson AFB OH, March Koutrouvelis, Toannis A. "Estimation of Location and Scale in Cauchy Distributions Using the Empirical Characteristic Function," Biometrica, 69: (1982). 20. Law, AverilU M. and M. David Kelton. Simulation Modeling & Analysis. New York: McGraw-Hill, Inc., Mendenhall, William and others. Mathematical Statistics With Applications. Boston: PWS-KENT Publishing Company, Meyer, Stuart L. Data Analysis For Scientists and Engineers. New York: John Wiley & Sons, Inc., Moore, Albert H. and Vincent C. Yen. "Modified Goodness of Fit Tests For the Cauchy Distribution." Accepted by IEEE Transactions in Reliability, Moore, David S. "Test of Chi-Squared Type." Goodness-of-Fit Techniques 3, edited by Michael A. Stephens and Ralph B. D'Agostino, New York: Marcel Dekker, Inc., Noreen, Eric W. Computer Intensive Methods For Testing Hypothesis. New York: John Wiley & Sons, Ocasio, Capt Frank. A Modified Kolmogorov-Simirnov, Anderson-Darling, and Cramer-von Mises Test For the Cauchy Distribution With Unknown Location and Scale Parameters. MS thesis, AFIT/ENG/GSO/85D-5, School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson AFB OH, December Rayner, J.C.W. and D.J. Best. Smooth Tests of Goodness of Fit. New York: Oxford University Press, Read, Timothy R. C. and Noel A. C. Cressie. Goodness-of-Fit Statistics For Discrete Multivariate Data. New York: Springer-Verlag, Ream, Capt Thomas J. A New Goodness of Fit Test For Normality With Men and Variance Unknown. MS thesis, AFIT/GOR/81D-9, School of Engineering, BIB-2

137 Air Force Institute of Technology (AU), Wright-Patterson AFB OH, December Ripley, Brian D. Stochastic Simulation. New York: John Willey & Sons, Inc, Shuster, Eugene F. "On the goodness-of-fit Problem For Continuous Symmetric Distributions," Journal of the American Statistical Association, 68: (September 1973). 32. Shuster, Eugene F. "Estimating the Distribution Function of a Symmetric Distribution," Biometrica, 62: (1975). 33. Sours, Capt John 0. A Comparison of Estimation Tachniques For the Two Parameter Cauchy Distribution. MS thesis, AFIT/ENG/GSO/MA/85-7, School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson AFB OH, December Spory, Ralph M. Conditional Best Linear Invariant Estimation of the Location and Scale Parameters of the Cauchy Distribution by the use of Order Statistics. MS thesis, AFIT/ENG/MATH/72-3, School of Engineering, Air Force Institute of Technology (AU), Wright-Patterson AFB OH, December Stephens, M. A. "EDF Statistics For Goodness of Fit and Some Comparisons," Journal of the American Statistical Association, 69: (September 1974). 36. Stephens, M. A. Tests of Fit for the Cauchy Distribution Based on the Empirical Distribution Function. Contract N J-1627, Stanford CA: Stanford University, December 1991 (NR ). 37. Stephens, M.A. and R.B. D'Agostino. Goodness-of-Fit Tecniques. New York: Marcel Decker, Inc., Weiss, G. and H.A. Howlader. "Linear Scale Estimation," Journal of Statistical Computer Simulation, 29: (1988). 39. Winkler, Robert L. and Williamri L. Hays. Statistics: Probability, Inference, and Decision. New York: Holt, Rinehart and Winston, Woodruff, B.W. and A.H. Moore. "Application of Goodness-of-Fit Tests in Reliability," Handbook of Statistics, 7: (1988). BIB-3

138 Appendix A. Computer Code For Critical Values A.1 FORTRAN Code for Critical Values of Reflected Tests CONNOR XX(10000),Pl~n1,RN,K1 INTEGER 111,12 REAL NEDIAN,NO,NSBO,BS,ZOZS,SLOPED,SLOPEV REAL MLEL,NLES,1(50000),Y(60000) REAL R(10000),DISA(100000),DISB(100000),PP(100002), 1 D(100000),V(100000) REAL ADO1,CNO1,ADOSCNOS,AD1O,CN1O,AD1S,CN15,AD2O,CN2O DOUBLE PRECISION DSEEDIU(10000) DSEEDI= ODO PRINT * ****CAUCHY REFLECTED CRITICAL VALUE TABLE **** REP=60000 PRINT *,',ith',rep,'replications' DO 10 I=1,REP+1 PP(I)=(I-.3)/(REP+.4) 10 CONTINUE DO =5,50,5 PRINT *,1' PRINT *,'For sample size N=',K1,' The CRITICAL values are' PRINT *,I'' 12=11*2 NNKX1/2 NR1=K1 SIZE=NR1*2 ADO 1=0 CKO1=0 ADO5=0 CN050~ AD 100 CN1O=0 AD15=0 CM15=0 AD20=0 CN20=0 CALL RNSET (DSEED1) DO 100 J=1,REP Generate the CAUCHY deviates CALL RICHY (TR1,1) DO 5 I=1,NR1 XX(I)=R(I)* ****Order the Variates *** NN=K1-1 DO 30 I=1,NN JN=K1-I DO 20 CONTINUE 30 CONTINUE 20 K=1,JM IFCXXCK).LT.XXCK+1)) GO TO 20 TENP=XX(K) XXCK)=XXCK+1) XI (K+1)=TENP IF (MODCNR1,2).EQ.0) TEEN XMED=(XXCNN+1)+XXCMN))/2.0 ELSE XMED=XX( (NR1+1)/2) ENDIF NEDIAN=XKED SENIQ=1O.0 ****Estimate the parameters****** CALL CNLE (NEDIAN, SENIQ, KLEL, NLES) ****Reflection about NLEL *** DO 32 I=1,11 Y(I)=XX(I)-MLEL 32 CONTINUE DO 34 I=1,K1 A-1

139 I(I)=Y(I)+NLELT 1(I+11)=-Y(I)+NLELT 34 CNIU DO 37 1=1, =IJ2 IF(X(K).LT.X(K+1)) GO TO 35 TEN=X(K) I(1)=z(K+1) X(K+1)=TEN DO 50 1=1,12 U(I)= *ATAN( (XCI)-KLEL)/NLES) IF (J.EQ.18) THEN ENDIF 50 CONTINUE DISE (J)=U(1) DISA (J)=( 1/SIZE)-U(I) DO 60 1=2,X2 ******Compute the distance from above(2)/below(1)****** D1=U(I)-( (1-1)/SIZE) IF (D1..GT.DISB(J)) DISB(J)=D1 D2=(I/SIZE)-U(I) IF (D2.GT.DISA(J)) DISA(J)=D2 60 CONTINUE IF (DISA(J).GT.DISB(J)) TEEN D(J)=DISA(J) ELSE D (3)=DISB (3) END IF V(J)=DISA(J)+DISB(J) ******Order The Edl Statistics *** NM=REP-1 DO 300 I=1,NN JM=REP-I DO 200 X=1,JM IF(D(1).LT.D(K+1)) GO TO 200 TENP=D () DCK)=D (X+1) D(K+1)=TEMP 200 CONTINUE 300 CONTINUE DO 305 I=1,NN JM=REP-I DO 205 1=1,JK IFCV(K).LT.V(K+1)) GO TO 205 TEMP=V (K) VCK)=V (X+1) V (1+1)=TEMP 205 CONTINUE ******Critical Value Computation For KS****** IF ((D(2)-D(1)).EQ.0.0) D(2)=D(2)* MO=(P(2-PP~))/D(2)-DC1)) BO=PPC1)-NO*D(1) Z0=(0.0-BO)/M0 IF (Z0.GE.0.0) THEN D(0)=ZO ELSE D(0)=O.0 END IF IF CCD(REP)-DCREP-1)).EQ.0.0) D(REP)=D(REP)* NS= CPP(REP)-PPCREP-1))/CD(REP)-D(REP-1)) BS=PP(REP-1)-NS*D(1) ZS=(1.0-BS)/NS DCREP+1)=ZS ******Critical Value Computation For KUIPER****** IF ((VC2)-V(i)).EQ.0.0) VC2)=V(2)* A-2

140 No=(PP(2)-PP(1)) /(V(2)-v(1)) BO=PP(1)-KO*V(1) ZO=(0.0-BO)/MO IF (ZO.GE.0.0) THU V ()Z ELSE VFO)=o.o UNDIF IF ((V(REP)-V(REP-1)).sq0.0) V(REP)=V(REP)*1.ooool NS=(PP(REP)-PP(REP-1))i(v.(REP)-V(REP-1)) DS=PP(REP-1)-NS*V(1) ZS=(1.0-BS)/MS V(REP+1)=ZS DO 410 P=80,96,5 DO 420 II=1,REP I=REP+ 1-II IF (PP(I).LT.(P/100.0)) THEE IF (D(I+1).EQ.D(I)) D(I+1)=D(I)* IF (V(I+1).EQ.V(I)) VCI+1)=V(I)*u1.oaooi SLOPED=(PPCI+1)-PP(I) )/(D(I+1)-D(I)) SLOPEV=(PP(I+1 -PPffI))/(v(I+1)-V(I)) ZD=-SLOPED*D(I +p(i ZV=-SLOPEV*V(I) +PP(I) PERD=( (P/100.0)-ZD)/SLOPED PERV= ((P/100.0) -ZV)/SLOPEV PRINT *,'The',P,'th percentile for D IS ',PERD,' GO TO 410 END IF 420 CONTINUE 410 CONTINUE DO 430 II=1,REP I=REP+ 1-II IF (PPCI).LT..99) THEN IF (D(I+1).EQ.D(I)) D(I+1)=DC1)* IF (V(I+i).EQ.V(I)) V(I+1)=V(I)*1.0oool GO TO 450 END IF 430 COTINUE 450 SLOPED=(PPCI+1)-PPCI))/CD(I+1)-D(I)) SLOPEV=(PP(I+1)-PP (I))/(V(I+1)-V(I)) ZD=-SLOPED*Df(I)+PP (I) ZV=-SLOPEV*V I) +PP(I) PERN. 99-ZD) /SLOPED PERV= C.99-ZV) /SLOPEV PRINT *,'The',99.,'th percentile for D IS 'IPERD,' 500 CONTINUE STOP END SUBROUTINE CNLE(NEDIAN,SEMIQ,MLEL,MLES) COMMON XXC10000),P1,R1,RN,K1 REAL MLEL,MLES,MEDIANMLELT,NLEST,MLESSQ MLEL=MEDIAN MLES=SEMIQ IMAX=100 ITER=0 40 NLELT=MLEL NLEST=MLES SUNO=0 SUM 1= 0 MLESSQ=MLES**2 DO 41 I=1,K1 Z=MLESSQ+ (XI(I) -MLEL)**2 SUM0=SUM0+1./ SUN1=SUM1+IXCI)/Z 41 CONTINUE TNLES=DFLOAT(K1 )/2.DO/SUMO/MLES** (1.5) MLES=TMLES**2 MLEL=SUM1/SUMO ITER=ITER+1 IF (ITER.GT.IMAX) GO TO 45 IF (ABS(MLEL-MLELT).GT..001*MLES) GO TO 40 IF (ABS(MLES-NLEST).GT..05*NLES) GO TO RETURN END for V',PERV for V2,PERV A-3

141 A.2 FORTRAN Code for Significance Levels of Sequential Tests COMMON 1(10000),P1.ffR1,RN,K1 INTEGER Nil,DD,VV,RDV,COL PARAMETER (ROU=20,COL=20) REAL NEDIANSEQ(1:ROV,1:COL) REAL RLELMLES REAL R(10000),DISA(50000),DISB(50000), 1 D(5O000),V(50000) DOUBLE PRECISION DSEEDI,DSEED2,U(10000) DSEEDI= ODO 1gtwD21H2146T.0D0 quniltest Program****** PRINT *,$**SEQUENTIAL TEST - KS(cols) and V(rous)**' DO 500 K1=6 110so' PRINT *, '****Sample Size 1=,K1, '****' NN=K1/2 111=11 SIZE=NR1 DO 1 DS=1,20 DO 2 VS=1,20 SEQ(VS,DS)0O 2 CONTINUE I CONTINUE CALL RNSET (DSEEDI) DO 200 J=1,REP ******Generate Cauchy deviates****** 15 CALL RICHY (NR1,R) DO 10 I=1,NRI X(I)=R(I)* NM=K1-1 DO 30 I=1,NM JH=11-I DO 20 K=1,JH IF(X(K).LT.XCK+i)) GO TO 20 TEMP=X (K) 20 CONTINUE 30 CONTINUE IF (NOD(NR1,2).EQ.0) TEEN XNED=(X(NN+1)+XCNN) )/2.0 ELSE XNED=X( (NI+1)/2) ENDIF MEDIAJ=XNED SENIQ=10.0 so CALL CMLE(NEDIAN,SEMIQ,NLEL,HLES) DO 50 I=1,Ki U(I)= *ATAN( (X(I)-NLEL)/NLES) CONTINUE DISBCJ)=CUC1)) DISA(J) =(1/SIZE-U(1)) DO 60 I=2,Kl Di=(U(I)-CI-1)/SIZE) IF (D1.GT.DISBCJ)) DISBCJ)=D1 D2=CI/SIZE)-UCI) IF CD2.GT.DISA(J)) DISA(J)=D2 60 CONTINUE IF CDISACJ).GT.DISB(J)) THEN DCJ)=DISA(J) ELSE A-4

142 D(J)=DISB(J) VE!JDISA(J) +DISB (J) IF (K1.EQ.5) GO TO 106 IF (11.EQ.1o) GO TO 110 IF (K1.EQ.16) GO TO 115 IF (K1.EQ.20) GO TO 120 IF (KI.EQ.25) GO TO 125 IF (K1.EQ.30) GO TO 130 IF (x1.eq.35) GO TO 135 IF (11.EQ.4o) GO TO 140 IF (K1.EQ.46) GO TO 145 IF (K1.EQ.6o) GO TO 160 **Critical value comparison for n=g*** 105 IF (D(J).LT ) DD=1 IF (D(J.LT.O ) DD=2 IF (D(J).LT.O ) DD=3 IF (D(J).LT ) DD=4 IF (D(J) LT ) DD=5 IF (D(J).LT.O ) DD=6 IF (D J.LT ) DD=7 IF (D(J).LT ) DD=8 IF (D(J)LT.O ) DD=9 IF CD(J) LT ) DD=10 IF (D(J) LT i) DD=11 IF (D(J) LT ) DD=12 IF (D(J).LT ) DD=13 IF (D(J).LT ) DD=14 IF (D(J).LT.O ) DD=15 IF (D(J) LT ) DD=16 IF (D(J).LT.O ) DD=17 IF (D(J)LT DD=18 IF (D(J)LT ) DD=19 IF (D(J).LT.o ) DD=20 IF (V(J).LT.O ) VV=1 IF (V(J).LT.O ) VV=2 IF (V(J).LT ) VV=3 IF (V(J).LT ) VV=4 IF (V(J)LT ) VV=5 IF (V(J).LT.O ) VV=6 IF (V (J.LT ) VV=7 IF CV J).LT ) VV=8 IF ýv(j).t..3314) VV=9 IF VJ).T.O39242)VV=10 IF (V(J).LT ) VV=li. IF (VCJ).LT ) VV=12 IF (V(J).LT ) VV=13 IF (V(J).LT.O ) VV=14 IF (V(J).LT.O ) VV=16 IF (V(J).LT.O ) VV=16 IF (V(3).LT ) VV=17 IF (V(J).LT.O ) VV=18 IF (V(J).LT ) VV=19 IF (V(J).LT ) VV=20 GO TO 100 **Critical value comparison for n=10*** 110 IF (DCJ).LT.o ) DD=1 IF (Df(J).LT.O ) DD=2 IF (D J LT.O ) DD=3 IF (D(J).LT.O ) DD=4 IF (D(J).LT ) DD=5 IF (D(J).LT.O ) DD=6 IF (D(J)LT.O ) DD=7 IF (D(J)LT ) DD=8 IF CD(J).LT ) DD=9 IF (D J3.LT ) DD=10 IF CDf(3).LT.o i) DD=11 IF (D J3.LT ) DD=12 IF (D(J).LT.o ) DD=13 A-5

143 IF (D(J).LT ) DD=14 IF (Dfi).LT ) DD=16 IF (D(.LT ) DD=16 IF (D(J).LT.O ) DD=17 IF (D(.LT.O ) DD=18 IF (D(J.LT.O ) DD=19 IF (D(J).LT.o.2o9504) DD=20 IF (V(J).LT.o ) VV=1 IF (V(J).LT ) VV=2 IF (V(J.LT.O ) VV=3 IF (V(J).LT.O ) VV=4 IF (V(J) LT.O ) VV=5 IF CV(J.LT.O ) VV=6 IF (V(J).LT ) VV=7 IF (V(J.LT.O ) VV=8 IF (V(J).LT.O ) VV=9 IF (V(J.LT.O ) VV=1O IF (V(J).LT.O ) VV=ii IF (V(J.LT ) VV=12 IF (Výi).LT.O ) VV=13 IF (V J) LT.O ) VV=14 IF (V(J) I.T.O ) VV=15 IF (V(J).LT.O ) VV=16 IF (V(J) LT.O ) VV=17 IF (V(J).LT.o ) VV=18 IF (V(J).LT.o ) VV=19 IF (V(J).LT.O ) VV=20 GO TO 100 **Critical. value comparison for n=15*** 115 IF (D(J).LT.o ) DD=1 IF (D(J).LT.O ) DD=2 IF (DC) JLT.O ) DD=3 IF (D(J).LT ) DD=4 IF (D(J) LT ) DD=5 IF (D(J) LT ) DD=6 IF (D(J).LT.O ) DD=7 IF (D(J).LT.o ) DD=8 IF (D(J).LT ) DD=9 IF (D(J) LT ) DD=10 IF (D(J).LT ) DD=11 IF (D J)).LT ) DD=12 IF (D(J).LT ) DD=13 IF (D(J) LT.O ) DD=14 IF (DC) JLT.O ) DD=16 IF (D(J).LT ) DD=16 IF (D(J).LT ) DD=17 IF (D(J).LT ) DD=18 IF (D(J)LT ) DD=19 IF CD(J).LT.O ) DD=20 IF (V(J).LT ) VV=1 IF CVCJ).LT.O ) VV=2 IF (V(J).LT ) VV=3 IF (V(J) LT ) VV=4 IF (V(J).LT.O ) VV=S IF C(J).LT ) VV=6 IF VJ).LT ) VV=7 IF (V(JU.T ) VV=8 IF (V(J).LT ) VV=9 IF (V(J.LT.O.26i920) VV=io IF (V J).LT ) VV=ii IF (VCJ).LT.o ) VV=12 IF CV(J).LT ) VV=13 IF (V(J).LT ) VV=14 IF (V(J) LT ) VV=15 IF (V(J).LT ) VV=16 IF (V(J).LT ) VV=17 IF (V(J) LT ) VV=18 IF (V(J).LT ) VV=19 IF (V(J) LT.O ) VV=20 GO TO 100 A-6

144 *** Critical value comparison for n=20*** 120 IF (DI).LT ) DD=1 IF (D().LT ) DD=2 IF (DCI.LT ) DD=3 IF (D(I).LT.O ) DD=4 IF (DI J.LT ) DD=S IF CD(S).LT T1) DD=6 IF (D(I.LT.O.18048) DD=7 IF (DC J.LT ) DD=8 IF CD(S).LT ) DD=9 IF (DCI.LT ) DD=10 IF (D(I).LT ) DD=11 IF (DCI J.LT ) DD=12 IF CD(S).LT ) DD=13 IF (DI J.LT ) DD=14 IF J(DCi).LT ) DD=15 IF CD().LT i DD=16 IF (DCI.LT ) DD=17 IF (DCI).LT ) DD=18 IF (DCI J.LT ) DD=19 IF (D(I).LT.O ) DD=20 IF (V(J).LT.o ) VV=i IF (1(3).LT )V= IF V11 J.LT ,i VV=V3 IF V11 J.LT ) VV=4 IF (1(1).LT ) V1=5 IF CVC J.LT ) 111=6 IF (11(1).LT ) V1=7 IF (VCI.LT.O ) 1111 IF (V(J).LT ) V11=9 IF V75).LT ) V11=10 IF (1(J).LT.o )1=1 IF (V(S).LT j VV1121= IF CV(J).LT ) 11=13 IF CV(I).LT ) VV1=14 IF C1(J).LT.o ) 111=15 IF (11(5).LT ) 111=18 IF (V15 J.LT.O ) V11=17 IF (11(1).LT ) 111=18 IF CV(J).LT.O ) 111=19 IF (ICS).LT.o ) 111=20 GO TO 100 **Critical value comparison f or n=25*** 125 IF (D(S).LT ) DD=1 IF CD(S).LT.0.18T080) DD=2 IF CD(S).LT.0.17'9512) DD=3 IF CD(I).LT ) DD=4 IF (D(I).LT ) DD=S IF (DCI.LT.O ) DD=6 IF CD(S).LT ) DD=7 IF (D(S).LT ) DD=8 IF (D(S).LT ) DD=9 IF (D J).LT ) DD=10 IF (D(J).LT DD=11 IF (DCI).LT ) DD=12 IF CD(S).LT ) DD=13 IF CD(S).LT ) DD=14 IF (DCJi).LT ) DD=15 IF (D(I).LT ) DD=16 IF CD(S).LT ) DD=17 IF CD(S).LT ) DD=18 IF (DCI).LT ) DD=19 IF (DCI).LT ) DD=20 IF (11(5).LT ) 117=1 IF V1(5.LT ) 11=2 IF (V(S).LT.o ) 11=3 IF V(VC).LT.o ) 111=4 IF C(S().LT.o ) 111=5 IF C(S().LT.O ) 111=8 IF CVCI).LT.o ) V1=7 A-7

145 IF JV(J).LT ) VV=8 IF (V(J).LT.O ) VV=9 IF V(J).LT ) VV=10 IF (V(J).LT )V=1 IF (V(J.LT.O ) VV=:121 1 IF (V(J.LT.O ) VV=13 IF (V(J).LT ) VV=14 IF (V8J)LT.O ) VV=15 IF (VJ).T ) VV=16 IF V(J) I.T.O VV=17 IF (V(JJ).LT.O ) VV=18 IF (V(J.LT.O ) VV=19 IF (V(J.LT.O VV=20 GO TO 100 **Critical value comparison for a-30*** 130 IF (D(J).LT.O ) DD=1 IF (D(J).LT.O ) DD=2 IF (D(J.LT ) DD=3 IFD(J).LT.O ) DD=4 IF (D(J).LT ) DD=S IF (D(J) LT ) DD=O IF (D(J).LT ) DD=7 IF CD(J).LT ) DD=8 IF (D(J).LT ) DD=9 IF (DJ).LT.O ) DD=10 IF (D(J).L.T ) DD=11 IF (D J).LT ) DD=12 IF D(J).LT ) DD=13 IF (D(J).LT.O ) DD=14 IF D(J) I.T ) DD=15 IF (D(J).L.T ) DD=16 IF D(J).LT.O ) DD=17 IF (D(J).LT.O ) DD=18 IF D(J) I.T.O ) DD=19 IF CD(J) I.T ) DD=20 IF (V(J).LT ) VV=1 IF (V(3) I.T.O ) VV=2 IF (1(J).LT ) VV=3 IF (V(J.LT.O ) 11=4 IF (1(J).LT ) VV=5 IF (1(J) LT.O ) 11=6 IF ýv 3).LT.O ) 11=7 IF V J) LT.O ) VV=8 IF V J) LT.O ) VV=9 IF (V(J).L.T.O ) V1=10 IF (V J).LT.O ) 11=11 IF ýv 3).LT.O ) 11=12 IF CV J) I.T ) 11=13 IF (V(J).LT.O ) VV=14 IF V(VJ).L.T.o.1846o1) VV=15 IF V(J).LT.O ) V1=16 IF ývfj).lt ) 11=17 IF V J).LT ) VV=18 IF (V(J) LT ) VV=19 IF (V(J).LT ) VV=20 GO TO 100 **Critical value comparison for n=35*** 135 IF (DCJ).LT ) DD=1 IF (D(J) LT.O ) DD=2 IF (DCJ).LT.O ) DD=3 IF (D(J) LT.O ) DD=4 IF (D(J).LT.O ) DD=5 IF (D(J) LT.O ) DD=6 IF (D(J).LT ) DD=7 IF D J).LT DD=8 IF (D(J.LT ) DD=9 IF (D(J).LT.O ) DD=10 IF (D J3 I.T ) DD=11 IF (D(J)LT.o ) DD=12 IF (DC(J).T ) DD=13 A-8

146 IF (D(J) LT.O ) DD=14 IF (D(J) LT i D0=16 IF (D J).LT ) DD=16 IF (D(3 JLT ) DD=17 IF (D J.LT ) DD=18 IF (D(J).LT ) DD=19 IF (DJ).LT ) DD=20 IF (V(J).LT ) VV=i IF (V(J).LT ) VV=2 IF (V(J)LT.O ) VV=3 IF (V(J).LT ) VV4 IF (V(J.LT.O VV5 IF (V J.LT ) VV=6 IF (V(J).LT ) VV=7 IF (V(J) LT ) VV=8 IF (V(J).LT ) VV=9 IF V(J).LT.O ) VV=10 IF (V(J).LT ) VV=1u IF V(J).LT.O ) VV=12 IF (V(J).LT ) V13 IF (V J.LT j VV=V:14 IF (V(J.LT.0.17i636) VV=16 IF (V(J).LT ) vv=io IF (V(J) LT ) VV=17 IF (V(J).LT ) VV=i8 IF V(J).LT ) VV=19 IF (V(J).LT ) VV=20 GO TO 100 **Critical value comparison for n=40*** 140 IF (D(J).LT ) DD=1 IF (DO).LT ) DD=2 IF (D(J) LT ) DD=3 IF (D(J).LT ) DD=4 IF D(J) I.T ) DD=S IF (D(J).LT ) DD=6 IF (D(J).LT ) DD=7 IF (D(J.LT.O ) DD=8 IF (D(J).LT ) DD=9 IF (D J.LT ) DD=10 IF (D(J).LT ) DD=11 IF (D(J) LT ) DD=12 IF (D(J).LT ) DD=13 IF (D(J) LT ) DD=14 IF (D(J).LT ) DD=15 IF (D(J).LT ) DD=16 IF (D(J).LT ) DD=1T IF (Df).LT DD=18 IF D(DJ.LT.0.1i1279) DD=19 IF (DCJ).LT ) DD=20 IF (V(J).LT ) VV=i IF (V(J.LT ) VV=2 IF (V(J).LT ) VV=3 IF (V( J).LT ) VV=4 IF (VCJ).LT ) VV=5 IF (VCJ).LT ) VV=6 IF (V(J).LT ) VV=7 IF (V J.LT ) VV=8 IF (V(J).LT ) 'JV=9 IF (Vffl.LT ) VV=io IF (V(3).LT ) VV=11 IF (V(J).LT ) VV=12 IF (V(J).LT ) VV=13 IF (V(J).LT ) 'JV=14 IF (V(J).LT ) VV=16 IF (V(J).LT ) VV=16 IF (VCJ).LT.O ) VV=17 IF V(VJ.LT ) VV=18 IF (V(ffl.LT VV=19 IF (V J) LT.O ) VV=20 GO TO 100 A-9

147 *** Critical value comparison for n=46*** 145 IF (D(J).LT ) DD=1 IF (D(J)j.LT ) DD=2 IF (D(J.LT.O ) DD=3 IF ~D(J).LT.O ) DD=4 IF D(J).LT.O ) DD=5i IF (D(J).LT.O ) DD=6 IF (D(J.LT.O ) DD=7 IF D(J).LT ) DD=8 IF (D(J).LT ) DD=9 IF (D J.LT ) DD=1O IF (D J).LT ) DD=11 IF (D( J LT.O ) DD=12 IF (D(J).LT.O ) DD=13 IF D(J).LT.O ) DD=14 IF (D(J).LT.O ) DD=15 IF (D(J.LT.O ) DD=16 IF (D J.LT ) DD=17 IF (D(J).LT ) DD=18 IF D(J).LT.O ) DD=19 IF (D(J).LT ) DD=20 IF CV(J).LT ) VV=1 IF (V(J).LT.O ) VV=2 IF (V(J.LT.O ) VV=3 IF (V(J).LT.O ) VV=4 IF (V(J).LT.O ) VV=6 IF (V(J.LT.O ) VV=8 IF (V(J).LT.O ) VV=7 IF (V(J) LT.O ) VV=8 IF (VJ).T ) VV=9 IF (V~ J.LT ) VV=1O IF (V(J).LT.O ) VV=ii IF (V J.LT.O i VV=12 IF V(J).LT ) VV=13 IF (V(J).LT.O ) VV=14 IF (V(J) LT ) VV=15 IF (V(J).LT.O ) VV=16 IF (V(J) LT.O ) VV=17 IF (V(J).LT.O ) VV=18 IF (V J) LT.O ) VV=19 IF (V(J.LT.O ) VV=20 GO TO 100 **Critical value comparison for n=6o*** 150 IF (D(J).LT ) DD=1 IF (D(J).LT.O ) DD=2 IF (D(J).LT ) DD=3 IF (D(J.LT ) DD=4 IF (D(J).LT ) DD=S IF (D(J) LT ) DD=6 IF (D(J).LT ) DD=7 IF (D(J) LT ) DD=8 IF (D(J).LT ) DD=9 IF (D(J).LT ) DD=1O IF (D(J).LT f7) DD=11 IF (D(J) LT ) DD=12 IF (D(J).LT ) DD=13 IF (D(J).LT.O ) DD=14 IF (D(J).LT ) DD-45 IF (D(J).LT ) DD=16 IF (D(J).LT ) DD=17 IF (D(3).L ) DD=18 IF D J).LT ) DD=ig IF (D(J).LT ) DD=20 IF (V(ffl.LT VV=1 IF (V( J.LT ) VV=2 IF (V(J).LT ) VV=3 IF (V(J).LT ) VV=4 IF (V(J).LT ?) VV=5 IF (V(J).LT ) VV=6 IF (V(J).LT ) VV=? A-10

148 IF (V(J).LT.0.164T42) VV=8 IF (V(J).LT ) VV=9 IF (V(J) LT.0.161T04) VV=10 IF (V(J).LT T) VV=11 IF (V~ J.LT.O ) V'V=12 IF (V~ J.LT.0.14T6TT) VV=13 IF (V(J).LT ) VV=14 IF (V(3).LT.O ) VV=16 IF (V(J).LT ) VV=16 IF (V(J) LT.O ) VV=17 IF (V(J).LT )vvi IF (V~i JLT ) vv=19ig IF (V~ J.LT.O ) VV=20 GO TO DO 101 DS=1,DD DO 102 VS=1,VV SEQ(VS,DS)=SEQ(VSDS) CONTINUE DI O 201 S=1,2 DO 202 VS=1,20 SEQ(VSDs)=1-(SEQ(VSDS)/REP) 202 CONTINUE 201 CONTINUE PRINT 400,((SEQ (ROCO), CO=1,COL),RO=1,ROW) 400 FORNAT(6(2X,20F6.5/) 600 COIU 1000 CORINU STOP END SUBROUTINE CMLE(NEDIAN,SENIQ,MLEL,MLES) COMMON X(10000) P1 NiR1,E,K1 REAL NLEL,NLES i~aeian,nlelt,mlestmlessq NLEL=KEDIAI NLES=SENIQ IMAX=100 ITER=0 40 KLELT=NLEL NLEST=MLES SUMO=0 sum1=0 MLESSQ=MLES**2 DO 41 I=1,K1 Z=NLESSQ+(X( I)-NLEL) **2 SUM0=sUMO+1./ SUm1=SUM1+x(I)/Z 41 CONTINUE TNLES=DFLOATCK1)/2.DO/SUMO/MLES**(1.5) MLES=TNLES**2 MLEL=SUN1/SUMO ITER=ITER+1 IF (ITER.GT.INAX) GO TO 45 IF (ABS(MLEL-M LEL T).GT..O01*NLEs) GO TO 40 IF (ABS(MLES-NLEST).GT..05*NLES) GO TO RETURN END A-11

149 Appendixc B. Computer Code For Power Studies B. I FORTRAN Code fot Power Study of Standard Tests COMMON x(10000),pi,nirnj.11 INTEGER RIl REAL 1(10000),DISA(5000).DISB (50000). 1 D(B0000),V(60000) REAL DO1,VO1,D05,V05,D1O,VIOD15,V1S,D20,V20 DOUBLE PRECISION DSEED1.DSEED2,U( 10000) PRINT *,)'******* power for dintributions **** REP 000 PRJ*, 'using' SlIP, 'replications' DO 1000 TYPB=1,6 IF (TYPE.EQ.1) PRINT *,'***PVR CAUCHY***' IF (TYPE.EQ.2) PRINT *,'***PWR NORNALe**' IF (TYPE.EQ.3) PRINT *,'***PWR EXPONENTIAL***) IF CTYPE.EQ.4) PRINT *,'***PWR BETA(3,3)***) IF (TYPE.EQ.5) PRINT *,'***PWR GAMMA***$ IF (TYPE.EQ.6) PRINT *,'***PVR UEIBULL***' DO 500 K1=S,50,6 M1=11/2 v01=0 V01=0 D05=0 D16=0 D10=0 D16=0 V15=O D20=0 V20=0 CALL RESET (DSEED1) DO 100 J=1,REP v* IhD a* j F~*j IFCTYPE.EQ.1) GO TO 1 IF(TYPE.EQ.2) GO TO 2 IF(TYPE.EQ.3) GO TO 3 IFCTYPE.EQ.4) GO TO 4 IF(TYPE.EQ.5) GO TO 5 IF(TYPE.EQ.6) GO TO 6 CALL RICHY (NR1,1) GO TO 15 2 CALL R1101 (nilr) GO TO 16 3 CALL RNEXP (111,R) GO TO 15 4 CALL RIBET (NR1,2.,3.lR) GO TO 15 6 CALL BEGAN (NR1,2.,l) GO TO 15 6 CALL RIVIB (111,3.5,1) 15 DO 10 1=1,111 x(i)=r(i)* DO 30 I=1,N1 JN=K1-I DO 20 K=1,JM IF(X(K).LT.X(K+i)) GO TO 20 TENP=X(K) B-1

150 I(1)=X(X+1) XXC+1)=TENP 20 COITIIU 30 COTIU IF (NOD(NRI,2).EQ.0) THE XNED=(X(NN+1)+X(NN))/2.0 ELSE XINED =X( (111+1)/2) ENDIF NEDIAN=XMED SENIQ=10.0 CALL CMLECNEDIAN,SENIQ,MLEL,MLES) DO 50 1=1,X1 U(I)=./, *ATAN((X(I)-MLEL)/NLES) DISBCJ)=(U(i)) DISACJ)=C1/SIZE-U(1)) DO 60 1=2,11 D1=(U(I)-(I-1)/SIZE) IF (D1.GT.DISBCJ)) DISB(J)=D1 D2=(I/SIZE)-U(I) IF (D2.GT.DISA(J)) DISA(J)=D2 60 CONTINUE IF (DISACJ).GT.DISB(J)) THEN DCJ)=DISA(J) ELSE DCJ)=DISBCJ) END IF V(J)=DISA(J)+DISB(J) IF ********** GO TO 10 * * ***** ** IF (K1.EQ.10) GO TO 1105 IF (Ki.EQ.10) GO TO 110 IF CKl.EQ.15) GO TO 115 IF (K1.EQ.20) GO TO 120 IF (K1.EQ.25) GO TO 125 IF (KI.EQ.36) GO TO 130 IF (K1.EQ.35) GO TO 135 IF (K1.EQ.40) GO TO 140 IF (K1.EQ.60) GO TO 150 **Critical value comparison f or n=6*** 105 IF (D(J).GT ) DOI=DOi+i IF (D(J).GT ) D05=D05+1 IF (D(J).GT ) D1O=D10+1 IF (D(J).GT ) D15=D15+l IF CDCJ).GT ) D20=D20+1 IF CV(J).GT ) V01=V01+1 IF (V(J).GT ) VOS=V05+1 IF (VCJ).GT ) V10=V10+1 IF CVCJ).GT ) V15=V15+1 IF (VCJ).GT ) V20=V20+1 GO TO 100 **Critical value comparison for n=10*** 110 IF (D(J).GT ) DO1=DO1+1 IF CDCJ).GT ) D05=D05+1 IF (D(J).GT ) D1O=DIO+1 IF CD(J).GT ) D15=D16+1 IF (D(J).GT ) D20=D20+1 IF (V(J).GT ) VO1=VO1+1 IF (V(J).GT ) VO6=Vos+1 IF (V(J).GT ) VIO=ViO+1 IF (V(J).GT ) VIS=VlS+l IF CV(J).GT ) V20=V20+1 GO TO 100 **Critical value comparison for n=15*** 116 IF (D(J).GT ) DO1=DO1+1 IF (D(J).GT ) DOS=D05+1 IF(D(J).GT ) D1O=D1O+1 B-2

151 IF (D(J).GT ) D15=D1S+1 IF (D(J).GT.O ) D20=D20+1 IF (V(J).GT.O ) VO1=Voi+i IF (1(J).GT.O ) V05=VOS+1 IF (V J.GT ) VIO=V1O+1 IF (V(J) GT ) 115=115+1 IF (V(J).GT.o.24356a) V20=V20+1 GO TO 100 **Critical value comparison for n=20*** 120 IF (D(J).GT ) DO1=DO1+1 IF (D(J.GT ) D05=D05+1 IF (D(J.GT.O ) D1O=D1O+1 IF (D(J).GT ) D15=D1S+1 IF (D(J) GT.0.iS3013) D20=D20+1 I F (1(J).GT ) 101=101+1 IF (V(J)GT.O ) V05=V05+l IF (1(J).T ) 110=110+1 IF V13 J.GT ) V15=V15+1 IF (1(J) GT ) V20=V20+1 GO TO 100 **Critical value comparison for n=2s*** 125 IF (D(J).GT.o ) DO1=DO1+1 IF (D(J).GT ) DO6=D05+1 IF (D(J) GT ) D1O=D1O+1 IF (D(J).GT ) D15=D15+1 IF (D(J).GT ) D20=D20+1 IF (V(J).GT.o ) V01=V01+1 IF (V(J).GT ) 105=V05+1 IF (V(J).GT ) 110=110+1 IF C1(J).GT ) V15=V15+1 IF (V(J).GT ) 120=120+1 GO TO 100 **Critical value comparison for n=30*** 130 IF (D(J).GT ) DO1=DO1+1 IF CD(J).GT ) DOS=DOS+1 IF (D(J).GT fi) D1O=D1O+1 IF (D(J).GT.o ) D16=D1S+1 IF (D(J).GT.o ) D20=D20+1 IF (1(J).GT ) 101=101+1 IF (V(J) GT ) 105=105+1 IF (1(J).T.o ) 110=110+1 IF V(IJ).GT ) 115=115+1 IF C1(J).GT.o i) 120=120+1 GO TO 100 **Critical value comparison for n=35*** 135 IF (DCJ).GT ) DO1=DO1+1 IF (DCJ).GT.o ) D05=D05+1 IF (D(J).GT ) D1O=D1O+1 IF (D(J).GT ) D15=D1S+1 IF (D(J) GT ) D20=D20+i IF (V(J).GT.o ) 101=101+1 IF (V(J).GT.o ) 105=105+1 IF (V(J).GT.o ) 110=110+1 IF (V(J).GT.O ) V15=115+1 IF (V(J).GT.o ) 120=120+1 GO TO 100 **Critical value comparison for n=40*** 140 IF CD(J).GT ) DO1=DO1+1 IF (D(J).GT ) D05=DOS+1 IF (D(J).GT ) D1O=D1O+1 IF (D(J).GT ) D15=D1S+1 IF (D(J).GT ) D20=D20+1 IF (V(J).GT ) 101=101+1 IF (V(J).GT ) 105=105+1 IF (V(J).GT ) V10=V10+1 IF (1(J).T ) 115=V15+1 IF (1(J).GT ) 120=120+1 GO TO 100 **Critical value comparison for n=45*** 145 IF (D(J).T ) DO1=DO1+1 IF (D(J).GT ) DOS=DOS+1 B-3

152 IF (D(J).GT ) DIO=D1O+1 IF (D(J).GT ) D16=D1S+1 IF (D J.GT ) D20=D20+1 I F (V(J).GT ) VOI=vo1+1 IF ~V J GT.O ) V05=V05+1 IF (V(J).GT.o ) V15=V1S+1 IF (V(J)GT.O ) V20=V20+1 GO TO 100 **Critical value comparison for n560*** 150 IF (D(J).GT ) DOI=DO1+1 IF (D(J.GT ) D05=D05+1 IF (D(J).GT ) D1O=D1O+1 IF (D(J.GT ) D15=D1S+1 IF (D(J).GT ) D20=D20+1 IF (V(J).GT ) VO1=VO1+1 IF (V(J).GT ) Vos=v05+1 I F (V(J)GT.O i V1O=V1O+1 IF (V(J).GT ) V15=V15+1 IF (V(J).GT ) V20=V20+1 GO TO CONTINUE PRINT *,'For Sample size',nfr1,l the power for D and VI PRINT *,'D ALPHA 1=',DO1/REP,' V ALPHA 1=',V01/REP PRINT *,'D ALPHA 6=',D05/REP,' V ALPHA 5=',VO6/REP PRINT *,'D ALPHA 10=',D1O/REP,' V ALPHA 10=',VIO/REP PRINT *e'd ALPHA 15=',D1S/REP,l V ALPHA 16=',V15/REP PRINT *,'D ALPHA 20=',D20/REP,' V ALPHA 20=',V20/REP 500 CONTINUE 1000 CONTINUE STOP END SUBROUTINE CNLE(NEDIAJ,SEMIQNLEL,NLES) CONMON X(10000),P1NR1,RJ,K1 REAL ELEL,ILES,NED IAN INLELT, LEST,NLESSQ MLEL=NEDIAN KLES=SEKIQ 40 Mt.ELT=NLEL HLEST=HLES SUM0=0 SUM 1=0 MLESSQ=NLES**2 DO 41 I=1,11 Z=MLESSQ+ CX(I) -MLEL) **2 StJNO=SUNO+1./ SUN1=SUN1+X(I)/Z 41 CONTINUE TKLES=DFLOAT(K1)/2.DO/SUMO/NLES**(1.5) NLES=TMLES**2 NLEL=SUM1/SUMO ITER=ITER+1 IF (ITER.GT.INAX) GO TO 45 IF (ABS(NLEL-NLELT).GT. 001*NLES) Go TO 40 IF CABS(KLES-NLEST).GT..05*MLES) Go TO RETURN END B-4

153 B.2 FORTRAN Code for Pow~er Study of Sequential Tests COMMION 1(10000).P1,NR1 RI.11 INTEGER NRI,CC,VV,ROVCOL PARAMETER (RDW=20,COL=20) REAL NEDIAN,SEQ(1:ROV,1:COL) REAL NLEL,KLESXI(5000O) Y(5i0000) REAL. 1(10000),DISA(50000),DISB(50000),C(50000), 1 D(5i0000).V(6O0O0),CV(60000) DOUBLE PRECISION DSEED1,DSEED2,U(10000),UC(10000) *****SeuenialTeat Power Program***~** PRINT *,'**SEQUENTIAL TEST-CV (cols) REFLECTED and V(rows)**' PRINT *,'Power of sequential test with',rep'1 replications' DO 1000 TYPE=1,6 IF CTYPE.EQ.1) PRINT *,'***PVR CAUCHY***' IF (TYPE.EQ.2) PRINT *,'***PVR NORMAL***' IF (TYPE.EQ.3) PRINT *,'***PWR EXPONENTIAL***) IF (TYPE.EQ.4) PRINT *,'***PVR BETA(3,3)***' IF (TYPE.EQ.5) PRINT *,'***PWR GAMMA***' IF (TYPE.EQ.6) PRINT *,'***PVR VEIBULL***' DO =5,50,5 PRINT *, '****Sample Size N='.11, '****' 12=11*2 111=11/2 NRl=KiM DO 1 DS=1,20 DO 2 VS=1,20 SEQ(VS,DS)=0 2 CONTINUE SCONTINUE CALL RNSET (DSEED1) DO 200 J=1,REP IF(TYPE.EQ.1) GO TO 3 IF(TYPE.EQ.2) GO TO 4 IF(TYPE.EQ.3) GO TO 5 IF(TYPE.EQ.4) GO TO 6 IF(TYPE.EQ.5) GO TO T IFCTYPE.EQ.6) GO TO 8 3 CALL RNCRY (NaIR) GO TO 15 4 CALL RNNOR CKR1.1) GO TO 15 5 CALL RJEXP (NR1,R) GO TO 15 6 CALL RNBET (NR1,2.,3.,R) GO TO 15 7 CALL RNGAN (NR1,2.,R) GO TO 15 8 CALL RNVIB CNR1,3.5,R) 1s DO 10 I=1,111 X(I)=R(I)* DO 30 I=1,NN JN=K1-I DO 20 K=1,JM IF(XCK).LT.XCK+1)) GO TO 20 TENPX ~K+1) X(K+1)=TEMP 20 CONTINUE 30 CONTINUE B-5

154 IF (MOD(N112).EQ.O) THEN XNED=(X(NN+1)+X(NN))/2.0 ELSE XNED=X( (111+1)12) lbf[n=xned SEI=10.0 CALL CMLE (MEDIAN, SENIQ, N.LEL. LES) DO 32 1=1,11 Y(I)=Z(I)-MRLEL 32 C8'BNTINUE +LE 11(I) =Y( I)+I UX(I+K1)=-Y(I)+NLEL 34 CONTINUE 1N2=K2-1 DO 37 1=1, =12-i DO 35 K=1,JX2 IF(IX(I).LT.XX(K+1)) GO TO 35 TEN=XX(K) NK)=1X(1+1) 35 CONTINUE 37 CONTINUE DO 50 I=1,K1 U(I)= *ATAN( CX(I)-MLEL)/NLES) DISB(J)=(U(i)) DISA(J)=C1/SIZE-U(1)) DO 60 1=2,11 D1=(U(I)-(I-I)/SIZE) IF (D1.GT.DISB(J)) DISB(J)=Dl D2=(I/SIZE)-U(I) IF (D2.GT.DISA(J)) DISA(J=D2 60 CONTINUE IF (DISA(J).GT.DISB(J)) THEN D(J)=DISA(J) ELSE D(J)=DISB(J) ENDIF V(J)=DISA(J)+DISB(J) DO 55 1=1,12 UcCI)= *ATAN( CzI(I)-NLEL)/NLES) 55 CONTINUE TEMP 1=12 WS=0. DO 70 1=1,12 G2=I C(I)=(UC(I)-(2.*G2-1.)/C2.*TEMP1))**2 US=WS+C(I) 70 CONTINUE CV(J)=1./(12.0*TENP1)+WS IF (K1.EQ.5) GO TO 105 IF (KI.EQ.10) GO TO 110 IF (K1.EQ.15) GO TO 115 IF (KI.EQ.20) GO TO 120 IF (K.EQ.25) GO TO 125 IF (11.EQ.30) GO TO 130 IF (K1.EQ.36) GO TO 135 IF (11.EQ.40) GO TO 140 IF (11.EQ.45) GO TO 145 IF (K1.EQ.50) GO TO 150 **Critical value comparison for n=4*** 105 IF (CV(J).LT ) CC=1 IF (CV (J).LT ) CC=2 IF (CV (3).LT ) CC=3 IF (CV(J) LT.O ) CC=4 IF (CV(J).LT ) CC=5i B-6

155 IF (CV(j) LT T8: IF (CVJ.LT.O.063TT95 CC7 CC=6 IF (CVJ) LT ) CC=8 IF (CVJ) LT )C= IF (CV(J) LT TI) CC=C:1O IF (CV1J).LT ) CC=11 IF (CV().LT ) CC=12 IF (CV(J).LT.O ) CC=13 IF (CV(J).LT.O ) CC=14 IF MCV().LT.O.046G730) CC=15 IF (CV(J).LT ) Cc=16 IF (CVGI.L B) CC=17 IF RCVJ) LT.O T CC=18 IF (CV(J).LT ) CC=19 IF (CV(J).LT.O ) CC=20 IF (V(J).LT ) VV=1 IF (V(3).LT.O ) VV=2 IF (V(J).LT ) VV=3 IF (V(J) LT.O ) VV=4 IF (V(J).LT ) IF (V(J) LT.O ) VV=8,r V= IF (V(J) LT.O ) VV7T IF (V(J).LT ) VV=8 IF (V(J) LT.O ) VV=9 IF (V(J).LT ) VV=1o IF (V(J)LT.O.3914T1) VV=11 IF (V(J).LT ) VV=12 IF (V(J).LT ) VV=13 IF MVJ.LT.O ) VV=14 IF (V(J).LT ) VV=i5 IF (V(J) LT ) VV=16 IF (VJ) LT.O ) VV=17 IF (V(JU.T.O ) VV=18 IF (V(J).LT.O ) VV=i9 IF (V(J) LT.O ) VV=20 Go TO 100 **Critical value cca)žarison for n=10*** 110 IF (CV(J).LT ) CC=1 IF (CV(J).LT ) CC=2 IF (CV(J).LT ) CC=3 IF (CV(J) LT.O.OT60680) CC=4 IF (CV(J).LT CC=5i IF (CV(J) LT.O.06T3681) CC=6 IF (CV().LT ) CC=7 IF (CVJ).LT ) CC=8 IF (CVCJ).LT ) CC=9 IF (CV(J).LT.O.oS74351) CC=10 IF (CV(J).LT.o.os54654) CC=1i IF (CV(J) LT ) CC=12 IF (CV().LT T) CC=13 IF (CV(JILT ) CC=14 IF (CV(J).LT ) CC=15 IF (CV(J).LT ) CC=16 IF (CV(J).LT ) CC=17 IF (CV(J).LT ) CC=18 IF (CV(J).LT.0.O453215) CC=19 IF (CV(J).LT ) CC=20 IF CV(J).LT ) VV=1 IF (V(J).LT ) VV=,2 IF V(J).LT.O ) VV=3 IF V(VJ).LT ) VV=4 IF (V(J).LT ) VV=5 IF (V(J).LT ) VV=6 IF (V(J).LT ) VV=7 IF (V(J).LT ) VV=8 IF (V(J).LT ) VV=9 IF V(VJ).LT ) VV=1O IF (VCJ).LT ) VV=11 IF (V(J).LT.O ) VV=12 IF (V(J).LT.O ) VV=13 B-7

156 IF (V(J).LT ) VV=i4 IF (V(J).LT ) VV=16 IF (V(J) LT.O ) VV=16 IF (V(J).LT ) VV=17 IF (V(J) LT.O ) VV=18 IF (V~i JLT.O ) VV=19 IF (V(J).LT ) VV=20 GO TO 100 **Critical value comparison for n=15*** 115 IF (CV(J).LT ) CC=i IF (CV(J) LT ) CC=2 IF (CV(J) LT ) CC=3 IF (CV(J).LT ) CC=4 IF (CV(J).LT.O.0T12666) CC=5i IF (CV(J).LT ) CC=6 IF (CV(J).LT.O ) CC=7 IF (CV(J).LT ) CCS8 IF (CV(J).LT.O ) CC=9 IF (CV(J).LT.O ) CC=10 IF (CV(J).LT.O ) CC=ii. IF (CV(J) LT ) CC=12 IF (CV(J).LT ) CC=13 IF (CV(J).LT.O ) CC=14 IF (CV(J).LT.O ) CC=15 IF (CV(J).LT ) CC=16 IF CV().T.O ) CC=17 IF (CVJ.L CC=18 IF (CCV(Vý).LT ) CC=19 IF (CVCJ).LT.O ) CC=20 IF (V(J).LT ) VVi1 IF (V(J).LT.O ) VV=2 IF (V(J) LT.O ) VV=3 IF (V(J).LT ) VV=4 IF (V(J).LT.O ) VV=6 IF (V(J) LT.O ) VV=6 IF (V(J).LT ) VV=7 IF (V(J).LT ) VV=8 IF (V(3).LT.0.264T57) VV=9 IF CVCJ) LT.O ) VV=1o IF (V(J).LT ) VV=11 IF (V(J).LT.O.26726i) VV=12 IF (V(J).LT.O ) VV=13 IF (V(J).LT.O ) VV=14 IF (V(J) LT ) VV=16 IF (V(J).LT ) VV=16 IF (VfJ).LT.o.248o89) VV=1T IF (V(J).LT.O ) VV=18 IF (V(J) LT.O ) VV=19 IF (V(J).LT ) VV=20 GO TO 100 **Critical value comparison for n=20*** 120 IF (CV(J).LT.o ) cc=i IF (CV(J).LT.o.o9o1711) CC=2 IF (CV(J).LT.O.o817736) CC=3 IF (CV(J).LT.o.o755807) CC=4 IF (CV(J).LT.O ) CC=5 IF (CV(J).LT ) CC=6 IF (CV(J).LT ) CC=7 IF (CV(J).LT ) CC=8 IF (CV(J).LT ) CC=9 IF (CV(J).LT ) CC=10 IF (CV(J).LT ) CC=11 IF (CV (J).LT ) CC=12 IF (CVf( J.LT ) CC=13 IF (CV (J).LT ) CC=14 IF (CV(J) LT.O ) CC=16 IF (CV (J).LT ) CC=16 IF (CV(J).LT ) CC=17 IF (CV ( J.LT ) CC=18 IF (CV(J).LT ) CC=19 B-8

157 IF (CV(J).LT ) CC=20 IF (V(J).LT.O ) 11=1 IF (V J.LT.O ) 11=2 IF (V(J) LT.0.2:=j43 IF (V(J) LT.O.2494)1= 11=3: IF (V(J) LT.O ) 11=5 IF (1J).lLT ) IF (V(J) LT ) 11=6 11=7 IF ((J.LT.O ) VV=8 IF (V(J) LT =9 IF (1(3).LT )110 IF (V(J) LT.O WV11=11 IF (V(J) LT.O ) V1=12 IF (1J LT ) VI=13 IF (V().T ) VV=14 IF ((J.LT ) 11=15 IF (V(JU.T ) 11=16 IF (1(J).LT ) 11=17 IF V1(J) LT ) 11=18 IF (1(J).LT ) 11=19 IF (V(J).LT ) 11=20 GO TO 100 Critical value comparison for n=25*** 125 IF (CV (3).LT ) CC=i. IF (CV J) LT.O ) CC=2 IF :C11:8:S ) IF (C(J.L ) CC=3 CC=4 IF (CV(J) i.lt.o ) CC=5 IF (CV(J).LT ) CC=6 IF (CV(J).LT.O ) CC=7 IF (CV(J).LT.O ) CC=8 IF (CV(J).LT ) CC=9 IF (C :)LT ) CC=10 IF (C 6).T0.5460B) CC=11 IF (CI(J).JLT ) CC=12 IF (CI(J).LT ) CC=13 IF (CV(J) LT.O ) CC=14 IF (CV(J).LT.O ) CC=16 IF (CV(J).LT ) CC=i8 IF (CV(J).LT ) CC=17 IF (CV(J).LT ) CC=18 IF (CV(J).LT ) CC=19 IF (CV(J).LT ) CC=20 IF (V(J).LT ) V1=1 IF (1(J).LT ) 11=2 IF (1(J) LT ) 11=3 IF (1(J).LT ) 11=4 IF (1(J).LT ) 11=5 IF (1(J).LT ) 11=6 IF (V(J) LT.O ) 11=7 IF (1(J).LT ) 11=8 IF (1(J).L ) VV=9 IF (V J).LT.O ) 11=10 IF (1(J).LT.O ) 11=11 IF (1(J) LT ) 11=12 IF (1(J).LT ) 11=13 IF (1(J).LT ) 11=14 IF 1().T ) 11=15 IF (1(~J) LT.: ) 11=16 IF V1(J).LT ) 11=17 IF (1 (J).LT ) 11=18 IF (V J).LT ) 11=19 IF (V(J).LT ) 11=20 GO TO 100 **Critical value comparison for n=30*** 130 IF (CV(J).LT ) CC=i IF (CV (3).LT ) CC=2 IF (CI(J).LT ) CC=3 IF (CV(J) LT ) CC=4 IF (CV(J).T ) CC=5i B-9

158 IF (CV(J).LT.o.oea1322) CC=6 IF (CV(J).LT.O ) CC=7 IF (CV J).LT.O ) CC=8 IF (CV(J).LT ) CC=9 IF (CV(J) LT.O ) CC=1O IF (CV(J~ LT.O ) CC=11 IF (CV(.J).LT.O ) CC=12 IF (CV(J) LT.O ) CC=13 IF (CV(J).LT ) CC=14 IF (CV(J).LT ) CC=16 IF (CV(J.L ) cc=1o IF MCIJ LT.O ) CC=17 IF (CV(J) LT.O Y CC=18 IF (CV(J).LT ) CC=19 IF (CV(J) U.LT ) CC=20 IF (Vý).LT ) VV=i IF (V J) LT.O ) VV=2 IF (V(J).LT.O J VV=3 IF (V(J) LT ) VV=4 IF (V(J).LT ) VV=5 IF (V J) LT.O ) VV=6 IF (V(J) LT.O ) VV=7 IF (V(J) LT.O ) VV=8 IF (V(J) LT.O ) VV=9 IF (V(J) LT.O ) VV=iO IF (V(J).LT.O.190S82) VV=11 IF (V(J).LT ) VV=12 IF (V(J) LT ) VV=13 IF (V(J).LT.o ) VV=14 IF (V(J).LT.O ) VV=i5 IF CV(J) LT.O ) VV=16 IF (V(J).LT.O ) VV=17 IF (V(J) LT ) VV=18 IF (V(J).LT.O ) VV=19 IF (V(J.LT.O ) VV=20 GO TO 100 **Critical value comparison f or n=36*** 135 IF (CV(J).LT.O ) CC=i IF (CV(J).LT ) CC=2 IF (CV(J).LT ) CC=3 IF (CV(J).LT ) CC=4 IF (CV(J).LT ) CC=S IF (CV(J).LT.O ) CC=6 IF (CV(J).LT.O.06S7629) CC=7 IF (CV(J).LT.O.o631797) CC=8 IF (CV(J).LT.O ) CC=9 IF (CV(J).LT ) CC=1O IF (CV(J).LT.O ) CC=11 IF (CV(J).LT ) CC=12 IF (CV(J).LT.O ) CC=13 IF (CV(J).LT ) CC=14 IF (CV(J).LT.O ) CC=16 IF CCVCJ) LT.O ) CC=16 IF (CV(J).LT.O ) CC=17 IF (CV(J).LT ) CC=18 IF (CV(J).LT.O ) CC=19 IF (CV(J).LT.O ) CC=20 IF (V(J).LT.O ) VV=1 IF (V(J) LT.O ) VV=2 IF (V(J).LT.o ) VV=3 IF (V(J) LT.O ) VV=4 IF (Vfj).LT ) VV=5 IF (V(J).LT ) VV=6 IF (V(J).LT.O ) VV=7 IF (V(J) LT.O ) VV=8 IF (V(J).LT.O ) VV=9 IF (V(J).LT.o.179o19) VV=10 IF (V(J).LT.O ) VV=1i IF (V(J) LT.O ) VV=12 IF (V(J).LT.o ) VV=13 B-10

159 IF (V(J).LT ) VV=14 IF (V(J).LT ) VV=15 IF (V(J.LT.O ) V1=16 IF (V(3).LT.01677) VV=17 IF (V~i JLT ) VV=18 IF (V(J) LT ) VV=19 IF (V(J).LT.o.1659o3) VV=20 GO TO 100 **Critical value comparison for n=40*** 140 IF (CV().T0172)Ci IF (CV (J L CC=2 IF (CV(J.LT ) CC=3 IF (CV(J).LT.O ) CC=4 IF (CV(J).LT ) CC=5 IF (CV(J).LT ) CC=6 IF (CV(J).LT ) CC=7 IF(V~i).LT: ) CC=8 IF (CV.T TJ CC=9 IF (CV(J) LT ) CC=10 IF (CV(i).LT ) CC=11 IF (CV(J).LT ) CC=12 IF (CV(J).LT ) CC=13 IF (CV(J).LT ) CC=14 IF (CV(J).LT ) CC=15 IF (CV(J) LT.O ) CC=16 IF (CV(J).LT ) CC=17 IF (CV(J).LT ) CC=18 IF (CV(J).LT ) CC=19 IF (CV(J).LT.o.o446692) CC=20 IF (V(J)J.LT ) VV=i IF ýv J).LT ) VV=2 IF V J) LT ) VV=3 IF (V(J).LT ) VV=4 IF (VJ).LT ) VV=5 IF (V(J).LT ) VV=6 IF (V(J).LT Q) VV=7 IF (V J).LT ) VV=8 IF (V(J).LT.O ) VV=9 IF (V J) LT ) VV=10 IF (V(J) LT ) VV=11 IF (V(J).T ) VV=12 IF (V(J).LT ) VV=13 IF (V(J~ LT ) VV=14 IF (V(J) LT ) VV=15 IF (V(J).LT ) VV=16 IF (V(J).LT ) VV=17 IF (V(3).LT ) VV=18 IF (V(J) LT ) VV=19 IF (V(J).LT ) VV=20 GO TO 100 **Critical value comparison for n=45*** 145 IF (CV(J).LT ) CC=1 IF (CVf(f).LT.O S) CC=2 IF (CV Ji.LT ) CC=3 IF (CV (J).LT.O ) CC=4 IF (CV (J).LT ) CC=5 IF (CV(J).LT ) CC=6 IF CCVCJ).LT ) CC=7 IF (CV(J).LT ) CC=8 IF (CVf( J.LT ) CC=9 IF (CV(3).LT ) CC=10 IF (CV(J).LT ) CC=11 IF (CV(J).LT ) CC=12 IF (CV (3).LT ) CC=13 IF (CV (3).LT ) CC=14 IF (CV(J).LT ) CC=15 IF CCVf( J.LT ) CC=16 IF CCV(J).LT.O ) CC=17 IF CCV (J).LT ) CC=18 IF (CV(CJ).LT.O.04S8S31) CC=19 B-li

160 IF (CV(J).LT.o ) CC=20 IF (V(J).LT.O ) VV=i IF (V(J.LT.O ) VV=2 IF (V(J).LT ) VV=3 IF (V(J) LT.O ) VV=4 IF (V(J) LT ) VV=6 IF (V(J).LT ) VV=6 IF (V(J)LT.O ) VV=7 IF (V(J).LT ) VV=a IF (V(J) LT ) VV=9 IF (V(J).LT ) vv=io IF (V(J) LT.O ) VV=11 IF (V(J) LT.O ) VV=12 IF (V(J).LT ) VV=13 IF (V(J) LT ) VV=14 IF (1(3).LT ) VV=15 IF (V(J) LT ) VV=16 IF (V(J).LT ) VV=17 IF (V(J) LT ) VV=18 IF (V(J).LT ) VV=19 IF (VJ.LT ) VV=20 GO TO 100 **Critical value comparison for n=60*** 150 IF (CV(J).LT ) CC=i IF (CV(J) LT ) CC=2 IF (V(J.L ) CC=3 IIF (CV(J).LT ) CC=4 IF (CV(J) LT ) CC=S IF (CV(J)LT ) CC=6 IF (CV(J).LT ) CC=7 IF (CV(J).LT ) CC=8 IF (CV(J).LT.O ) CC=9 IF CV().T ) CC=10 IF (CV).L CC=11 IF (CV(VJ).LT.O ) CC=12 IF (CV(J).LT ) CC=13 IF (CV(J) LT ) CC=14 IF (CV(J).LT ) CC=15 IF (CV(J).LT ) CC=16 IF (CV(J).LT ) CC=17 IF (CV(J).LT ) CC=18 IF (CV(J).LT ) CC=19 IF (CV(J).LT ) CC=20 IF (V(J).LT ) VV=i IF (VfJ).LT ) VV=2 IF (V J)LT ) VV=3 IF (V(J).LT.O ) VV=4 IF V1(J) LT ) VV=6 IF (V(J).LT ) VV=6 IF ((3).LT.O ) VV=7 IF (V(J ILT ) VV=8 IF CV(J) LT ) V1=9 IF (V(J).T ) 11=10 IF (V(J).LT.O.15o167) VV=11 IF (V(J).LT ) VV=12 IF (V(J).LT ) VV=13 IF (V(J).LT ) VV=14 IF (V(J).LT ) VV=15 IF (V(J) LT.O =16 IF (V(J).LT.O ) VV=17 IF (V(J).LT.O ) VV=18 IF (V(J).LT ) VV=19 IF (V(J).LT.O.14o647) VV=20 GO TO DO 101 DS=1,CC DO 102 VS=1 VV SEQ(VS, DS)=SEQ(VS,DS) CONTINUE 101 CONTINUE 200 CONTINUE DO 201 DS=1,20 B-12

161 DO 202 VS=1, COT 0TIBUE PRINT 400.((SEQ (RD,Co), CO=1,COL).R0=1,1L3V) 400 FORMAT(5(21,20FG.S/)) t880 ~88191 SUEROUTINE CKLE(NEDIAN,SENIQ IMLEL,HLES) COMMON X(10000),P1NR1,RN,K1 REAL NLELINLS,NEDIAN,NLELTMLEST,NLESSQ INAX= iaft==zl.r N T=NLES KMUSQ=NLES**2 DO 41 1=1,11 Z=NLESSQ+ (X(I)-MLEL) **2 SUNO=SUMO+1./ SUN1=suN1+x(I)/Z 41 CONTINUE TNLES=DFLOAT(K1)/2.DO/SUMO/MLES**(1.5) NLES=TMLES**2 KLEL=SUN1/SUKO ITER=ITER+l IF (ITER.GT.INAI) GO TO 45 IF (ABS(MLEL-MLELT).GT..01*MLES) GO TO 40 IF (ABS(NLES-MLEST).GT..05*NLES) GO TO RETURN END B-13

162 Appendix C. Probability Points C.A Probability Points of KS and V Tests C-1

163 I Probaility Point for n = 51 Probaility Point. for n = I 1 V II 1- as IS v W W ' C-2

164 IProbaility Points for n =10 Poalty Points for ni = L10 f--a s 1-a j R Js 1 KV v V & , U ,, , C-3

165 Probaility Points for n = 15 robilty Points for n-= a [ $ I V 1-0 KS [ v & E E E & & O.M & & E E E C-4

166 I Poitforn201 Probaility Point forn= 20 1-,-. K si I '-az [ s I v U7, M O E E F, S & & & & E E & E E E C-5

167 I Probaility Points for n = 25 F Pro;ity Poi, for n = I xks v '-a [ I KS v & & SE & F, & , & & _ E E & E F & & & E ,768658& F & E E E E II0,I C-6

168 Prn Pons p typos n s.9amf-, ] F, & & & & E F & & & , & & & & & & & & E E F E & & F & V' & & E E & & E E E E & E E & & & C-7

169 D I PP Proat Pont KS for Eý n =W 31 [[ýt 11 Pont fok r n =_!!] U PP KS -- F v I W& , &, S F E P F & E & & F E E , & E E & & W & & & & & E E ] E E & E & & & E A E E E E & E C-8

170 Foity PoUforn 40 ý y Poitsfor n 40 PP S Is H PP KS I vs , & & P, o & E, & & & F F, E & & E & E E E E E E E E E F E , E E E & ] E E E E & E E E E E E E & M & E & E E E & , E E C-9

171 I erobtilty Points for n = 45 1 F P-oty;ut, Poi r = 45 1 PP if I "y 11 PP KS V _ & M & & Z & F & & & F & W F & & & F, & & , & & & & S & & & E E & E & & F F, E E E & E E & & & E f, ý',; E i E & E E E E & E F, C-10

172 I PoPsf5 P. f. 0 1 i o ty Po,or n 60 15o IZIv" PP KS I V PP KS , :2-8, & F, & & & E E = & F ] F Z & & E & E M & & E & & & & & & TO7-O & F & F & & & & & & W & & E E F & & & & E E & & E & & & & & E & F E E F E & E E & E E E & C-1l

173 C.2 Probability Points of CM and CM(Ref) C-12

174 FProbWbity Points of Standard CM Test]I )I n=5 n10 n= 15 n =20 [ n= M Probability Point. of Standard CM TestI (I 1 -)II n 30 n = 35 n = 40,n = 45,,n= C-13

175 FProbability Points of Reflacted CM Teatl nii~ & & & S & r & F, & & & & E & & & E & & & & S & & M& & & & &02~ P & E E E & & F, & & & & & & F, E & & F, , F, E F, =5 n= 10,=15 -T =2 30 n= 25 ] biit Points_ of Refeced f )l - n= 0n=3 n=45 7 n =45 n= & E E E & & F, i E F, E & E E & & & & F, & E, & T ý F, F, & M j F, & E E , & F, E F, F, C-14

176 Appendix D. Power tables of CM - V This appendix includes the complete results of CM - V Tequential Test. The tables includes the power levels of the test against the Cauchy, Normal, Exponential, Beta, Gamma and Weibull respectively. For each alternative sample sizes n5(5), 50 is covered. After the tables for each alternative distributions the corresponding power graphs are presented. In the graphs "o" represents the power level of V test and "*" represents the power level of CM test. The straight line "_' represents the power of CM - V sequential test. D-1

177 1 ~ ~ a.a. ~ ~~R Iiwa.a a a.. 40, 4ma -.aa. M* a.... a ~ ~ ~ 41 c!.. q~- -.. w I aaaa-aaý eca..;aa. - qa v 7,.rp-3o r 3 1.U Oe.. a~.... a-ae.q -15 P-61.a *0-0w CC 26 S 2 C41 IaC.....a. e......la..a 0~... 'p0 go. a ea. n a. a ~~~~~~0. *eaa -. ae - a -0. a.. F! nr o a e aa e-11f "T16 ' P oi -o e e aa e a 1.. n mso aaaaaaaaaa.!ii ta r I.a-a.. Ialfa m a.- C O 0... a.. ~ a cc,o e O o 6 na. 10,a-. Xor R ci CoCOe0 0 r--i ri -2 r D12 2

178 -~lb - *4l0W'~*I ' ~ * A ~ A10~ ~ l~ on~fq *.10.. n ~. e 0000-i.a0. *.e4 - ~ a 0 ~4. W0..n W.~. 0 f. 0 *.* n... PinW~*~* ~f 0 ob.18"s 2 o owe owloo l. 0. Cal-e--2Z-n-----AfAia fl *. * w l* n o w 0,0.0 wi t *. 0 :0 a~~~~~~~~ wo e o w *.en...0O lot "V 1a w" wow 0o-*04 e. no. 1S fl " R eo.. w n 2ow w e no C;....0l. o ~ ~ - ~ O ~ o ~ n ~~C Wn 00 C.0w w I0.O..w wn. n CO. w.,- 1.0 w n.. oftl~f.o~inn... wo nn W o.-. E nnnino... I. 4 C4 0n fi~~~ oow n ri oonnoci I I-- Ii O ~ * Wfl 0 wl0f a...ea.0o ~..-- o lllff -~ ~~~~ ~ e. n n 2nn w l o w w. o e w w - f o o 0 0 bn n n 00 eq n n 0o0l.ff l - ~ ~ ~ ~ ~ ~ ~ V D-3!r: I-

179 ;.- :e Zn a-. e.. n. n a e 0 ~P III~nCC#.@@U 0= II~~ t...c.....n ". n n CFlC C* 0. A.11 Z IS....cnc..#-. q. -.. n., ~ 0.n~nr"qn"n - - o o.. c e.u. n... n..... a 1 X C l-x t a -CC 0 n S -! eonfn"nc."inc-f m onb A e T. nni c -"n ICn, 2 *~Ir X P *,1; -W 14f0 I. 0 C t e" nn 41 A 0n "~e~ A C! e! -CV!.Cl r! ~ a ~, 0 In00-0 I'm 0a 0 a - *. IV. 0 *.~nne on~e m. on.u V!C C. 4CO SSt! * *C b-~0..0lbc@ ewe-. ~ ~~~ 0*. * 0 0 I 0V 0 - o %, e: - ~ - ~ ~ - 26 n i C o -O~ e - ISo1oeO 10,eSO 3 a 0 v0 00e10,oI a"n... VDn.C. CO! " oo i. D

180 -~ ~~ ~~~n 0. (QN * - *QNNeefl-~.@@~N0N -1 * 0 *en 14 A 0.0 a.0 fl 0 ** NU's~l......N..e.NN N. A n aub.0~ I Ibf~b@ N 0 E O N A o ~ ~ ~ a 01:0i*~~ ~ ~ i o EN..a 0 X r #7 2 ;r~~n a ar N * ~~~ ~ ~ CN.0.NO NO~ O ~~ r~ N~ *UO.0 1N~.NOtO o ;.NN0@.NN~.*-~. OW~O~N A 0 O..OU".O N -Nr-ON 0, r, 0 e " O N- : N.O.. YU 0 ~ C, COCCOCqOnC UP IN0 ad-5a

181 ril 3~ 210Ca s, - #n a.~00ei o.o.. on P1 114~ NN ;- O.. i P 4i! V ~1 o *oo eo a-oe vo0 a IS.XnIV 94 it~~~~. ~~~~~ O On 10f 2 l ~ I* f00l~~ C ~ ~ 0 n 0 W CC 0 2 fa 0 C U -0 : = - = 0foff *~~~~~~~~~ uf nn1.e.fll ZOfffO. o 1o4.~ 20 S f le- 000%~ 1" e -. 0W a O ;00 * 0.S 1i iq V! v! i t gi A 23 0 ffll 2rl W f0 l-fls00 ra 0-0 ; 1-11 r;10 P-! ---, P0if0 I x fl * ~ ~ ~ ifff 0~ ~~~~~~1 10f~ 110l 10 0 no = 04oo~-.-w aogn Z I o v.01-. a o.~n n -; q -if 22=3~0~ 00 o~910.. ~~io0 10-n R 1* *. s ý:. C I : D-a

182 - ~~ ~ 1 :11@~l0*w. - WN.. Ar 0l :.@.o. p *.* PO@0.. e-e o a c; n 1! 00 n0e~~ ve- nnc q~a 0? ni. 0 X.ao., a * O0=0 *.0 - o; 0Oe. q 0 *0C1. n. a0....~.e... *~.#..a *~~ o~ V 0 "Sqee X~ E~~E~.. 'a~ a. o..f~@..u...a.."d - o. oe o w a n qo WW E0~~~ n no 21 Y.01 #@ **@ n.l04 a ~Ew-0~ 0~W0 ~ D-7

183 "A 1 1~ " 00 12gf~ a l a 0 a, c; e. o. a c.a.. te~-ae'k. 3 P cooat 0e.t Is 0 1*- "moaa C,~~ 0-Nt A zp Pig, r.. g c;i, ~~o o e o, 0..-.n ~ ~ 0 30 'go 0 eqna0a aa o a.w~eq~a..w qeo. m 1~- -t f0 t." *00 *. oý a.eqaa A.o xo onna 0 0 ý~~qeet " 2., i OP 00S "0 a.0 0 a Owwon..n~eaawnO.-n; "O. OO A.0 i~ Ca -q ao -. e.w..... f0~ prna.- W. ;Pr S 0 0e C,.10 to vd-81 B

184 -0 12i eq.1 2 0" x vw s0 MOrit a n 'I w INa. 0. bq.1 e0o a"9x w 2 a. 00w 2.q 20 Xq - ae e X e e a e ww 2 e e NO e.. N I a 3 114,60 1 "Cl11 T 0 * ~~~ ft 4%fn w ee. W -W 01 w. ~ q.. a ~ -... e eeo cqweainoeaqeq0%ve o.e q w ooees w C.qqeq w ~oe. ~ eqee -..e q N P 0.1e.10 N 2 a.10 toq we* 3We U~0e 0 eq~c.~~u0* qc ~ os.# 101, ow wee~~a e re02, o 1w0 z.o. mq~qe sw eo~~o 0 o en.u ooeeee w.oo. 600qU a eq~~~~~~~~~~ e~oe w q w *s eq0 zo0u e q ee..e Vq.w- q 0~~~ ee4vdneeqa-cew 8U1000*U 10. C., 0 0.0eqnneqeeeq 0 wmnn qwweee 0 V e Z~ :11 0OS" o4 Ct D-9 n-x

185 o-l A-Pir: ~ -PO P.z1:2 "o' 0 ;6 oo 0 0 * - ~.o~ ow.c w~-.. U 0 ~ne.aee no~. eq. C E*~*~@l n on e ~~~~I!It' ~. i. IoqA ~ c; 141 o.cq~0 E *6OM l8' qq SO 9.U1 Pn 20 00U PV. :!" C Q e. i.. C.04_V A ~~~~~.10 a e e o m M.0.020f~.1 W.. 0 OC4.w*O e n -! is ~.0 * 410 E 3.)C a. 0 Ce. n- o! 0C lq~~~c 0. q 2 0 ~ 0 on~il: U-08 S.a a zr 9 "0 USURZOVWER MW -Ma a 2 An '00 0", ~ agcl ~ ~ RFEC).0"T *,"M On D-It

186 .OI ~~~~r ;Fqe c; - oo o e n..i.-i ;0 0 0 f P0-O 3 0p loop.0..04e o oa 10-00" f ~ N 6.e eqi b. ~ o F r 0 t c;a 1,oooooa ~0... e0..ee~..02. o o oo g e e e o o o O ~l 0 o 01non v0 ne o a e o o,.g.o ~ a o o eq. 00 o on no..neq n n~e#oq3o - 4om 9 :o o n ~ o n o G!oo 4! C,,0.wobo c'.00 noo-aaoooo a~~~~~~~~~~~~0.a-o 4"o.oo~eo q~a00nq~- ~ * 0~~~ 000~e 00e * M C; ý.0 le.080 VoD.11 15

187 Power of CM-V against NORMAL for n= "'.PwofCM. pe of Seq CM-V Significance Levels Figure D.1 Powercoprss of GM against -V Normalor =5 0-1

188 Power of CM-V against NORMAL for n=5 Power of CM-V against NORMAL for n= Significance Levels Significance Levels Power of CM-V against NORMAL for n=15 Power of CM-V against NORMAL for n= , "r Significance Levels Significance Levels Figure D.1 (Continued) D-13

189 Power of CM-V against NORMAL for n=30 Power of CM-V against NORMAL for n= O Significance Levels Significance Levels Power of CM-V against NORMAL for n=40 Power of CM-V against NORMAL for n= , Significance Levels Significance Levels Figure D.1 (Continued) D-14

190 2 r"..z MU Mf N V A X asola * *n.*n~ * *~~ aa aa a aa a a P. ;12 10~ - a ccc.a v x ~ ~ ~ ~ ~ P IV Inq~qf~qtqqc ac I~~ r c cc w: 1.. oýo P P Pc P cl CX ccivs ~ a 9i *..ea. i - i a...o r-! I' P IO " 1---aa.0.Pac9A.. c; roeci eb. o 0 0sO.eabC I,. Cc C! Ci I Ci C; A X.& 2 0 0e- It.. i eai c..e W ;~ N-; % n ""no C, 3010 'il no12d-.1

191 0 a. % 1ý 26 'a a o 2 IM o0de.. ~ * B * a-0 OW4 WI W 0 U o 0 f*wwcq0 0flI~ U b a! 0 0 'o- WOOO* fw I IW Co C I ~V P*0 r! P.-, 2 1.lo.i pi o. iw a s O so O W f W *'.. I. i I.. I. I *I e n a'... e o Cl~~ W~ CS ":!.-0aC.0 C -1tO a0 0 Cs 0 0CC la. sw1 0 0 w A-' 2- "- * s t C W C' C-C00a0oCSs0 O W! 0!aS C00-C s~C0 0sC In0 ~ C0~ W 0 C~~e 0 ~ ls, a" 1- a-* 2s0, a00cfw 0-o.ss 0a00C~W ~ -0.. Pm-16

192 o ~ ~.*e. ~ ~ ~ ~ 2.* ~ ~ * a OV 0 *a* *O* Pq ar :p Peq P. 10 v 04 St.o~q.., a"eq,.~ 2 a~~ *0* ~ T.4** * a~~~~".0e# 0 0 4!e0U~00ae C; ý11 Pil Fg.4l0. 00 P - *F-- a ag soqooeewea~~. o0 0 ono. 0 : ea!a~e.o 4 a a.1a 0 f ea fuov *M~q~ae, o ae. a 0.. I. 0 C D311

193 H E *~ * * ** * * * 0 0 * a..~..e.e c.0. ~. c. ~ e ~ ( ~ ' e 44e C4! 4! 4!*~ M :44 all ill! all ~ ae, ~ o. 0 I.... a a4 a a Q.6~ a. aq a0l4 a. a... ( 4 *Q... we*..**,*~ *.~~ ~... ~...0. * *.0,~ 4 4 I.. lilu o.n o.:.oo..cq~~:.~ 0,04C~0*~04~@'a * ~ ~ ~ I 0b Pa******~ a :0 l ** T SO "lo0 OPN M. 'a0~ 0.0 U 04 * C,. a.0ai A " all M 14@ 0.14 * ~ O C,~~ C, I ,W 44* C!4! Vo a!, ; t.- moia "a 11: : 0:: 0~ a a a a 404 a a a04 0.~~ ,~ 1 1,~ 0C,~00 11,0,6, a-60~ a a a. C, 0 ol 1 Io0.,41 'o@n6.s v O * " A :I" 'a 0 ~.4 C,.1 0w 1.0@C1, 01,1, ell,0a., In 6 0 0!4 a o.o. C, C, X 40 o.. 00 cq A D

194 "A*Vmet ~P P 0.0.)) 0...** 0..t a)..0 O )000C0~~-*C I CC o;w IN00 C) ~ ~ C ' ! P0.10 P00 6 4! d!@00 4! G! a. tr oooooooooo.0~ ;1; Ile w % ~ ~ ~ CC)0 ~ ~ C ~ ~ 0 ; 'so C00 00O 01: )).00 C M. o.. C, M Z f 0 N ar e.o o :10 o o o o 20o o oo1",o1o el 0001 c;.0..0 a & I00.. or me ::0.. C :::a::: -C-0 C, 00)0CC b 'A 000 )0 ) CC00 CC )C0 0C000)))00 C; C C) 0000 C C) C) 000 CC)0C.0C ~ 11 rc000-e00.0c) I` I0 P00 IC-0c. f!i 9.CC0 C0000)0 0 a )O CC0) 0. G! M woo 0 4! 10 a 0!0CU0 Cf! -- *0 - ) 4A)00a C le.0 e0im* M T ao - a1 q-1

195 i Power of.i CM-V against EXPONENTIAL... for n=25 ii i. A.. "!D: pw Of V 0' i !......*: pw of CM: *.;_.: pe of Seq CM-V I 0.1 I 0.15 I 0.2 I Significance Levels v I Power of CM-V against EXPONENTIAL for n= :;... o s M v... *~P. pof CM L:. pe of Seq CM-V Significance Levels Figure D.2 Power comparisons of CM - V against Exponential D-20

196 Power of CM-V against EXP for n=5 Power of CM-V against EXP for n= '' Significance Levels Significance Levels Power of CM-V against EXP for n=15 Power of CM-V against EXP for n= o i... 0o i " Significance Levels Significance Levels Figure D.2 (Continued) D-21

197 Power of CM-V against EXP for n=30 Power of CM-V against EXP for n= !~ Significance Levels Significance Levels Power of CM-V against EXP for n=40 Power of CM-V against EXP for n= ) "... 4) Significance Levels Significance Levels Figure D.2 (Continued) D-22

198 .01.00~.,n a a. 0 I C,. A A * 00 %f.0 r-.! 12o-.. r.cl 4Csoo. ; I soso b esor 0 6r. Pss- 1;4r-s PT I e w l l igw~s~q ! "n..oc, Z ; r. Io. rooq Ies. -. 1!3s.oc,", I s 1. o 1. nso so.s..@cs 2o -oo.. sr ~ al. o o "2Z #os I ai o..ss o.oss 4, s 9Csswo IX.C1 os"19 so~c ls l@ l-o O I IV o C,woo mo R e l. A.I-. l-. ~~~~~~~~~~~~ a. so c nlxrk- o so1.s.-% e os ;sso a-w". o s Olip RI.~ W-Os o o~ :=,"WHIM:.C o o~ lo o *C s so o..-. soscc, V. me 10 :rso Aso-. va~~. le -0 " eq ~ ~ ~ ~. a.o-soc :s~oc~-o 11 weqoqs0c oo.l- -o 0 cc.* --. s MRogong ; Ice "M o tooa~ C.w 00, a"w so~so~oeqs C-.o~lVso w -e IQ..Cs *CsesFC -~ ~~f -- l-~ss~eo-q -as~ssewqq-ee-see. rm m"m -1" m0 c 0c

199 ftfttft 1 X. IV a a ::- vv -! :A... 2ft" ~ wt*.,e a. * OAS0.0W06 o ef W f.. Is X 0~~~~C 0@f.0O1ft ftf.ft ft 'Ca.*0 10 t 0 0. o a A a2 r 19! a 2 ~ ~ *f f 1W * W 000* 11 0 C, I's.~t 00.1f 0 ~.00 * Sa f.0 S a..f COD. ~.. ~ ~ ~ ~.f0 0f 0f0.tf 0. Pill 1.0f.0t.0f.0 ft.00f0.0b UtfO 10f ft0 ~ t U~ ~ fffffo.. o.. 0U. U Z; za 1-~ft so -' ;O0t~tU 000; f=C.09: ; * ei; ;P q. o Otf ft0.u " U...0OfU0t i i 0 fftf0f ; ft.0tu f eft.-.. U.. O t-0t.i PfO I I O f.0 ~ t.t a.0 m ft ti.0ru It0P.0 I I V.... f~....0ft... 0a fo. f~..rl r.ui0r. -~~~~T a0 00.ftffttft000 UU ~ s0 0 ~fftfft00u UU ag ; 21 ;Pr.0~t~tUt.eUU R C, , 110, C eoooooaoo 0UUUU~tttU '19-2Y

200 'Aun eo *.0 e a - a.... e 1!n a-oe.e nn..2s I 5,* n ~ n ~ sa 1.0 e ca e 0. a-n. RE.n-ma- nn a-e 4, 14"m2 n Pit I...I..' a1 0 at ni tl au 0, 1 IlF,1II 1 It Iq b t11 al 71r-I a. *~~~~~~~ onixu~~ N.a~u. Ii P ru-awi'lle..n Siga. %11.*.O 9.@ R X pie a * uu~ao e~so~ o~~~~~ q ~. on C.nuamea--a... 11~m.0nou C Cv IOnuas. a-... :. ne10ces o~~ ".n-,o~oo-~n * N S nw aeta- a '..... r...1 Sna V4 -va-n... aa- I., -. Ci.n O.5a-.C a-. n,. -. 5O.-.n.e.. 2%ma V!I.n.a-u~~na-owe~~nmae..o~~ 0nnmmeaa numaa-. nue e- 0 ee ~ -~ Mea.n u e-.0 g. II,; ; r 0 D-V25 I II I IQ

201 'o.e**~ 21;., 1 ; Io F- riap rat P e.. Vo !!! - R.. oos*q a! q 4! 0ie! ~ e.6 Iv RVn*oeno *..soos~o. W.020 ~.i 0*0 *i Is o~~~~~~~~~~~~ *oc4sseqo ft-s..s. Cti~- EU~( -~~~~= -0!i o-noo-,s0o 0 A10 00V0n o.n~fea-fls,-.fls~soflfflc~o flw rc - vo (i.( n I a "..o v X a n " ; ;~ P ; :O "Qfoso a = so ees4c4qsoso..oo1oo'-e (is~ so. b- To 4, 12.o,o 0. o b os s a o s a m s A obo#a-aesesosossos1 oi* s. *.o sosossosos 0 sops - 0.Ossosos..,-O o v.a.s 1-0 c; IV*oossuosoo es ams os.bOs... t~s 0!....e s s.. 1-! 0!d!G D-I-"

202 ai a o ~*. ** PIP~ 41 I o 12 " AM *** 33 o r e... o o U o *UoIfeCU.. o "! 0W.~.~*.... *... a oo.. o 01C,F1 2 4! G! o sa!o 4 G N.~ nt... (. a... ~ o W W W * W W - -e,.-w 0 ItU W UI I ~ 5 0 : 0.I. 0c : 0 A* a.2 0 o~~~ 4! D-2

203 I Power of CM-V against BETA for n= ii li i.. S0.6 0 pe of Seq CM-V Significance Levels Power of CM-V agaist BETA for n= A... S0.4 i......!.. o..f *. ': pwofcm: i o s i u v I I III, L:. pe of Seq CM-V Significance Levels Figure D.3 Power comparisons of CM - V against Beta D-28

204 Power of CM-V against BETA for n=5 Power of CM-V against BETA for n= S ' ' Significance Levels Significance Levels Power of CM-V against BETA for n=15 Power of CM-V against BETA for n=20 1 1, " Significance Levels Significance Levels Figure D.3 (Continued) D-29

205 Power of CM-V against BETA for n=30 Power of CM-V against BETA for n=35 1,,,1 r Significance Levels Significance Levels Power of CM-V against BETA for n=40 Power of CM-V against BETA for n=45 0.8, o Significance Levels Significance Levels Figure D.3 (Continued) D-30

206 A A..IIi..I.II i ifp I-.*p 1I'lr if c; E "W ~~~~ I~ W-1sosso....hw 9.ns ecn.o. -. ~ ~ ~ ~~ -o ~~~~- - ~.dbn ITssn~C~ee...Wb 1 ~ ~ 2 SOf0* *.t~~ O *0*~~ ~~~~~ ov I.P- I- as. Cal@,*O w QsS 1.1 o c o20 n1 oso 0q, a z 13riri R1 11i Poo~oso r.csa I' r4 P*q r- e C - r. so.. *...- ~~sosooa-eeo1.0.s c s o S. e. c s e e. s a. ~ q o o - ~ ~ ~ sl 1.. ~ 10CG o * nsoso sooo..;ooe~-o '0 "'" n - - " " V. C;.0s*... so Ss OSi U' U4-0 C; 0 l.r " OP P-0,. ossb.ooeoc ooo., ;r lb~ qq P so ~~so1 sooss~ooooq soosso ro o. rc 0 10 o~~~~~~~~~~~~~~~ sososos-s C.ci 1oas.e...sos so C - s o-cd w o o a Rsss.,o-coooCss I; rqse.*ssfqsea.oq o~~~~~~~~~ soso P.ssooocssnooio-o. ~~~~ P- 1sseoss..os2s. I-- oso- ;I; zr* "D, :So C xooz Z s so o. o~ ~ ~ ~ ~ ~ r IsS.s~loo~-Uss.ss I I III I I r*o..sososo... sososo~~~2 ;;1" 1 IN~nooo~.sss.10* 1oS tso o. e. o C so... o. so o....s Csae.q.. a.,s... o. o. o sos o s..... s.sfo oc sss0...es D-19P

207 *~~.j'8~.q'8'8'on* 30 8.,. i V I *0t8 b It*88 It~n I*** n 1.1 II- a'8 ~ I ~~.'10 *8@ I 0 lop0~' * ripi -- ;1. -7I q1-.1;r- -- -n r-q 1- P r;~' *13Prr. 1-1 P. P- r7p1;p r;'c# 1;11 ; r7!,...' P - ; e- "... *e 1 P I' --t P ; Is P I. rv- Pe= C. 1. CI, aa O" *1,E : e.-.- w. o..... : W I1.' 8C 88f 4 fl.c~p.;;. P P8C8 r;-e 'I- i, T T;.CP Ir "I a q n fin'.0c; 00 f8 ~ '' 4 S ~ 8 It..'C888l.4.0 ~~~i ~ ii i i l It I.' qnini q" "I loq W A O'so 0 0) OVS lv 2 X88~'''~888~' noq..8 I8, "SI C4 W W C0 '8 f. O~ ~~ ~ P''''~8'' Pq~l'' PC 188tC' 0..'' f ''I ' I -. ~~~ 8 C ~ ~ ~.'* ''C'* " 8 ' ** ~ t. ' C. C- a1-'88'88 z '8 1C -.n IcO xp.l ,.11'8' k- Cfl INtl2681; P 26a Xt"I' "Ise I,., I 0 aon. *-ft w.4 w :&-Z g,: Zg,D-,.

208 N 1 Z '~~na 00 OV ~ a.e0. V. 4ee v *~~~....5~.. ~.l.o. B B I~~~ ~ P 21 r ;WI P1Cf~.ElB.CEf P ; ic*l all~b * 5,' Wl.U~BBCB~.4~ 1 l 14~ P ;! P - B P P 13P ae- P -1 a-ba **aaae - BW.0 Is Is ~ o * a 0. lb~cb Fa lb f II 0~1 a ~- 77 V a.. l N~qaeea- 0 RR".. :p 5 B S BCC1 0'C1S *B BBc;0B a -. *.BiBl*. BC... 6 It. W CB ClB Pi LIB *a9* BBB 4 6- F ' - BB. a a- a.6. 'a. I- C p a a - 0D CE. '1a 'a~abb B a U ~~I f** I** ~aae D-3

209 o 0 d...;.p!.oi 1.. ii a,. oce. eail.ja e -~ ~ ~~~~ s j b n e n eb C A r* a. p~ *.0b les a a a. 00 O.1100 a Z 4! 0! 4! 4! 0! 4! 4!** C; e. nct M n CotnA n. C; At O A j 0~ : n. a s an 0 0 V3 3am 0. ~ no ~ A 1 n am nam& e~. 0.a n e.e.e.ea n n. ACA~~~~~~~' e'pr.(jsn ~e AAOOOO00 It c; n n- e 1A. 2 P -e P.A P;O.. qwnc; o. so n n " 11an ae. n 4!..ne.nn.n.A.e4! -~~~~~~.ncn -ee o ~ 1.0n1.eo.. v- o -0 ofu OA6., AA A f A * ~ n a C ~ o oe Anft neo f~eo q... en.. a nn A U~ b-ac fjc b-cýi i G!...n oc! 4! 4!e n oooooi-g! g-e n~~~~0 P...n -PC~wnC ~ or ewe eq n nne nn n n*cb-.e.an.eno C gee7ne1n OeAeC Q-SSUOOOCOIA C ~ ~ ~ ~ C o ~ ~ ~ ~ ~ ~ ~ ~ ~ C sn n e~.... n e~obo C.e A s- 20. *;00 A ear! ebea..m 0, en,. eeeo!~ D-No-

210 a::: Pt.I. ipp-2 PPO 2l P FE O EE.0@E WIG 41 E C;#-~O -- - I: ; m- p ::o... P 11 Q W # C E veq W.1 v~0 f 0 - C~ ~~ ~ -~~~~' * * SMECE a0c a I1W 4, * 0 * 4.01 n..0qe0i.oka 1 70 r:- P 071 P I P I. r; IIP o -P Zo "or... 9EE* r~o ZPe@@ W CEE b~..@ 1E 1E E0a0 SO* U0101 "ac. 26EE E. CE0106C01CWOC)E- CE40cWCC C W C 0 H o.0.w.ee.c.c.c.c1c 1E--ECC C C C CC ~. ~ i r.0.!00 r.8 P11W1 W00.C EC ~. ~ ~ ~ CCC1C 1" 10 W,;sa a-d"o20ar

211 Power of CM-V against GAMMA for n= Q..!..., " I..... O ' 04 - : D: pw Of V 0 :pw of CM pe of Seq CM-V Significance Levels 0 Power of CM-V agairnst GAMMA for n= o i S i....:pw of CM pe of Seq CM-V Significance Levels Figure D.4 Power comparisons of CM - V against Gamma D-36

212 Power of CM-V against GAMMA for n=5 Power of CM-V against GAMMA for n=10 1 I! " Significance Levels Significance Levels Power of CM-V against GAMMA for n=15 Power of CM-V against GAMMA for n= i ' I Significance Levels Significance Levels Figure D.4 (Continued) D-37

213 Power of CM-V against GAMMA for n=30 Power of CM-V against GAMMA for n= of Significance Levels Significance Levels Power of CM-V against GAMMA for n=40 Power of CM-V against GAMMA for n= S I 0.4 t Significance Levels Significance Levels Figure D.4 (Continued) D-38

214 41,1 10 * c;e - 1 z P 0. e ne na p. -e o a In ~~~1 s -T!nwa P! I 1.na- ~ e a a e aw * n ft w 010q Viw 'a w V wee a aa ew, c0 ~ n. fl '0 :0 v,"-ee.1 12w...8.oca 0.W ws X- e c.e01 w1.,;r aa n Os.fne % ew aan..0 a 1aca aae n.c w.. c n 4e. ~~~~~ M c. ea.a,w -a. c! 0we~-nw -an ee na e.,a,a..wea.aw. coo D-399

215 P6IS 1 ~ 4. 00:: C,. 4 6 ~ r; P66 P~ 04 P P- P P. ~ ~ ~ ~ 6 ~ 64 ~ ~ ~ C 9:?" P--66 P? I! rt 6 P-O6 P6646 Porp6-qP B. ~ ~ ~ ~ ~ ~ C *.*~ " t 0o q B 0 4 m 0 6 f666 6 Va e 6 C1 6 ~ 4%.0f 6 a W * 'o W a 06 4I m 6 4 no4 0 M"6 1~062 6ow 46vb-a 0;M.k 0 O66 "Vl6 W. t no 610~6 00, 1.0 lo F 6.0 F4 I I I Ci V?4 1! 66 6 C6 64 ran p-o v o a T b 1.11, 0 0,I @ ~44666b b-6 ~ ~ D-0

216 0 ; -ri rigq* r- p.. - N!. I ~~~~~~~C a 0:0~o.. en.. D a 1. N on O I I I I '! n.a....a. oq * **0 C4.. qa0 1C006 3v Im.0 0I~.. *. r.0 c;~na~0 * 00i1a o 0 P. r; Pa~tif -~U x 0 0 I' iitc Zm IV ur,..a@ ~ U 0 P. a- aniq.~e... laoa a.0 C; l osi *--aain i i i Iq is.. I* 00@~i~~ a 26 nin aa-~~~ia.-e XiI;aR D-41b

217 ci I-- p P 11 p 11; 1 ri 11 ri 11 P 114. =W12 I i -! i f i Oil! i F to 0 6 '"s IV F F 01, r 'no OUT I It 0: 0 N S 0.0 v no.19, ffe P p P igr a Z;; A C; "A.0.0 I P F--t rl r; 13 r; P JS.6 F P It" ;Por ipf;.1 p I P! I A I '00 a, C.- IS 0 2" ft me v C ell. im"s"os: z 3 lap ;p 310, x v gý a,. r 26 fte.0 Pr IPF U _lg ýj s V I Z. I.. 11 NI: Mosr 10, X v N.C. 100 a a c; 2.9 a P r-4 P 1 P, ,,D;g rzl A ; "Wo f m So C, SAO 10-0 a W ; P P P.; `1 111 r ;is 1121 E.11 a V0 ID 0 Ir MoP 000. Ow 0 ""VW:00 a cc cc c; ;ýc; c; c; c; C,1010 s c; 0 c; c; c; c; D-42

218 a 3-. Q~a ~l a a. ~ oa~a~~ ~u~ a ~ so. ai 3 a C.n 00 Bat-on....,.o~ o.... q...~.. a.. *t 4! 4!G! a!.d ~..... o ; i: 2a ~ :~ A Ae~~ 2i0 a o...a o o.. o... a. a a~....e.- "o gigaaa a,.,~~~ o 3a.a--r :Pr 9%; em.a a. oa.a.aaoo ano.. a an.- n a~n a.0.. C; onaata a -1. 4!a a Q! d! -a a a a -a 00na a-.o.a el W! 1 -oo-q* 2.,oaaaatot- N aaon aoan 00-V-rBan a oa..an -e-a-a. -a- a- - d e aa- aaaa oaoawaa an...a V! q W!"q a a ~..*e.. oo.. oe.. lp D-43

219 Power of CM-V against WEIBULL for n= I pe of Seq CM-V Significance Levels 16 Power of CM-V against WEIBULL for n of.seq..m-v Significance Levels Figure D.5 Power comparisons of CM - V against Weibull D-44

220 Power of CM-V against WEIBULL for n=5 Power of CM-V against WEIBULL for n= " o,0 O ' Significance Levels Significance Levels Power of CM-V against WEIBULL for n=15 Power of CM-V against WEIBULL for n=20 1,, 1,,, O Significance Levels Significance Levels Figure D.5 (Continued) D-45

221 Power of CM-V against WEIBULL for n=30 Power of CM-V against WEIBULL for n= I ' I ~ S Significance Levels Significance Levels Power of CM-V against WEIBULL for n=40 Power of CM-V against WEIBULL for n= , S Significance Levels Significance Levels Figure D.5 (Continued) D-46

222 Appendix E. Power tables of CM(Ref) - V This appendix include the complete power results of the CM(Ref) - V Sequential Test. The results are presented as tables and garphs for each alternative distribution. On the graphs "*" represents the power level of the CM(Ref) test and straight line "-_" represents the power of CM(Ref) - V sequential test. "o" again represents the power of the V test. E-1

223 a 4CA so O~iCCC ON *...C-k CC C. CC C C O I I ~ ae CC "nn~c. - w eb C IU a V! 11 n 0011 CC! Me U-. CCCC~ b-s..flc CWC I Mo C C CC C C C S 6a.C rc 11.. CbCCrIC C rci C 1 0 C. CU VI-C C C U - S.. q. C C C C C C U OC CV C V0 ACU C~ ~~~~~~.. ~ ~ C ~ ~ C ~~ ~ ~ C C. C C ~ I U C C C S C - C C - C CCC A.. C C@COIS.C #UC 'o CC.CCC -0.I.I.C St b-~bccciclawcc"'..cc 4 ow.... C.CC V!b-C14?C1 1C 1!C 1 C1 V! C.CCitCon..CC.I.I..q.- 40bC.-.q.e.i. C. E C CICC -C@ SCLCICC acc C C V C C I CIC C CC.1C10C Pb5I CCIC t C' o~~~~ I C C I U C - C C C q C C * C C O C C O ~ ~ ~ C ~ l ~ b* ~ ~ ~ CC p II ;CC 9C- ; C. R ;r E~fo R o" C... I... CICC C (U* -OC.LUC C LUCI U C CC -*CI U-C CCCC.CR `FCEt CC C10 C L C C L ~ 0 C z CACC C x.e-2... Ot 0--

224 ft -Iq~f~t ff~q ~ f.f.t ff tt ff* * *~~~ tff *t@ ~ MY ffft* PI ~ ~ v " i ; n.ft* a... t t bffbft "a.e *tf~o. ftm 41 a a Nq - t o ~ e t f~fff0 0.4fwtfW 0t W W ~ ~ c "lon a*t~l 0.f A.0ff~tt 24fO 0t~t f@ t ar "I- 5.~ o. t..o D~oo ge e 5. a*2:2 f 00 : f f ftto a owoffftffff0 o0 v 6. a. O O l 0flt~ 0; ft0 ff H IPft 2t. t f t c 2t - - t 0 'Sh~~ f 0 f ~. St t 6tt t.t " t 34 tt. X ff 0 f ftoftft ~ -0f0 ~ ~ ~ ~ ~ tlfo ~ ~ O w w!to O fffffff or DO 'ft 1 P-0 1;f P't tt0 0 0f..ff 'Jf5fC~0.f Pttttt0W O Offffff ft~~~~~~ qt f~tf~tt- ft Vtt0t~ w a 2 a a a a a a a '4 tf 0 ~ 0fvtlff0 ~ ~ O-tt Of3f w v ct5oa-t.0zft0f 0A ftftfttlftf I**ff~f5 "a ov 'Iwo 2 afftfftt ft ~c 0 0 vffff0 5f n, 00 f 5- : &. lol 0 " 7,voo~oo oo E-3

225 a- v Z S2 g.0 00 a a.. e q..~u. 0. f.~te. u ~..u. *~ ~~~~~~~~ oa.ww. a u... qwe...~o 0 u. ~ ~ a a. a r! rl..u~~~ea.e P F.P P- * *a0f0.ue0eq0e.8 9 C Atw uor Z Sa' ;I 0 C XC; olo O wo JU41

226 - I- -.10a ~ **Ol~.0 0 a ec;1. : I 0 (tc-.0 *0Cl;~ 6.~ lf ItA 31! U t1l W U O l C 01100O a Old,. 0.1 **Cl0~~~~~0,0l0.@ C000 b-0 1 0l l.. a r..*-.--.-,a -... *0@ Z 00o0C lC " l01 It a0 Ole.ea a l a oo4...a - 0 a.!0 o ~~0 111Ul * ~o 0C e p.~oloc I16l10.1. a O ~ 00 IF0-.I..l00 i P ;.0C~ ;0 11 r am0Cl 01 00lIe C4 v 11 v M O ~0 Cll1C0000 no 0l 04 00I0---C0C S 2 e000 e 0 2-1

227 *00....ow.. q a.. i... o a.a. a 0a00 ~.. a.0 a * q t. e. q a n. a. a-a..r.. a.. n '2e a n..o... qan o; I 0. 0"~ "a 20 #.,**.0s~ 2 fa *0 mo f~ *. to-..o.1n -a a a n on.. e -. a n 8e a...o.n Ise. a a a ~ C C uaaa P P117~annna i 0l e ac,.c C Co pin. d!oqa a"n 0 a".e eeoc!a!.. a o.0 a *.aa. o.1n.. oq.n Ce.n 4! er! 0. 0 a. e n e!a a ~~~. C! a. PU a i -s -... C!!! d 4!4!. Wt. J! M e O P Ir-,, I`r- PP. ri-6

228 Power of CM(Ref)-V againal NORMAL for n=25 ~ M~ef... A b:pe of SeVMrt- Power~ ~ ~~~~P ofo(e)vagi OMLfo n=50d e.mre p0. of" Significance Levels Figure E.1 Powercoprsn of CM(Ref) - V against Normalo n-5 I E-7

229 Power of CM(Ref)-V against NORMAL for n=5 Power of CM(Ref)-V against NORMAL for n=10 1, ". " S06. t i...i o i... : S Significance Levels Significance Levels Power of CM(Ret)-V against NORMAL for n=15 Power of CM(Ref)-V against NORMAL for n= Significance Levels Significance Levels Figure E.1 (Continued) E-8

230 Power of CM(Ref)-V against NORMAL for n=30 Power of CM(Ref)-V against NORMAL for n= / Significance Levels Significance Levels Power of CM(Ret)-V against NORMAL for n=40 Power of CM(Ref)-V against NORMAL for n=45 I I! , o o Significance Levels Significance Levels Figure E.1 (Continued) E-9

231 *~~. In.1f ~ ft 2 wi~ a ~ I2 a. we n ft ICI wo a eq M;e a e... - ~x. '~(I2~ ema --,, 3. 1f "so as" 0.00"@me aaw.*, a~ w...,t..aa ine.. oa Na r- ni r- p Ir -- m.. 0,0.e Aee~. w.1- c o~.0.em,0 m e Os V a a Z-1O

232 BUSO 6 -.ee u iv w00 00, 60 5 F.0023 I n* e~ to Sa 3 Oýi * 1-0 P P0b-0O.1 o ~ ~ ~ e -.0.0o..o a 0 ~ t0 616*.l oe0 ee 0* eqe.q. o..0 q0. q a @ eq.. eq e o0 Iw I- 6 00, - 0 ~ r I.q0.0w.q..n re.w~ 0 eqeq 1 r rrowe Ow. o ~ ~ q0 ; P Z a.c w C;1 w001 ~.01e0 1* w 100..w O661eq e o.6e w w I.~.O o ~ ~ I 'a 1., :22: X e we e. o~. w.. ft e a C; 000ft "~ -0 w - q.o - o 0.r O 06 O O lo C I. a." c e a 09 a 2a2 2 1; 04 1; ý ri rzt-re-111

233 00 b e ~ -*~*o e. e 0...,o e ;..0 ~ ~ tit !0 f. I00..0.@ Z...0. ~~ ~ ~~~~ ~mem 000 e : , 0. 0 b o0 2; O 2I e @ b C 0. c; ,0t 1 X I' I.0 1~ V0 q w ,001, 00 Sola - C 1, 1, C 10, 1" 0 a" Ica I 10 a I I v v ! ! 4! 0! 0! 4! 4! 4! G! Is U l A.00a.P0 0 0 :0."a a 20 ao a" N 0*"'D @@ V.* Za CC :2 at 4! v go t 0: 1 t 4t r I I r0 0 o *0.0, 0 ".0 0 " "M 0 Wý 2~O =29=2.&-0 O~O~C@ O~q.0.0.8t..0C; 0.0 0@e.- 0 q P.~~~~ v sCq a.c " 1.00, U000~ IV Z Cc, SIP% IR A " q l@ I.M CC" fl00m."m I.- aa ~ ft 00A0a0"0a01. " C.C ~~~~U e IZOIN X A P11 V 1 i!xe-1 1 gr2n;99 ; Z ;; ;

234 40-3.,... Ao 0 ~. ~ 2 ::::-~ 2 Aa c;.0 b.o !.A t ri OO 0 000,5 tq 4!.4!.0!O.00w aw 4OV0j b V0 5l00 * A. 0 f *~ 0 C** 6.** 00 00*0* '.... *"a**r;*p. 0 *- *0.0 e. 00sw0r 11-o % 0 b.. a A000a I I t I a 45 I's 00 4! 0 l I n0 5 r I 111"0 -P A M C, 1 - ai- = *~~~~~~ ow~a5a5 00 ~ 1- o oo M. M:ooo 0.o. n.55 A o oo o 0! N 0Sa aa a aa a a0i 0 v0 o oo o o~ a.o o.ooa.00 n 51- a 0 w , I:::::,00~ A0 5 't a I 0 00 a 4! ~ 0 ~ ! %,5,,00, r I r PI-1,00E-1 r

235 1: ri It. P.1 Il 2. Sa s;x. a 1 :1 III 2~~4 ** * * *00 -~*~ G! 4!! 4! 4! 0 al a *. e. 4!!... a... a.e...., o 2 2Z :2" co on I ~ a m-.. ot 00" A. ~ ~...~0e 0 0~ a @ a. 101,a 0a a a 0 St a.me's 000: a a m.0 Noa0-l0I.00;0;; :10 0 * 00 soo ::::o;:z:2 2:2: 0 2 i?2~~ *.. a* 4! I!4! 4!4!4! uc; C92:22200@a a 0Ea

236 Power of CM(Ref)-V against EXPONENTIAL for wn=25 * ~~pe of SqCMiref) Significance Levels Power of CM(Ref)-V p against EXPONENTIAL for n ~ o: pwof V ~ pw of CM~ref) pe of Seq CM(ret)-V II 0* Significance Levels Figure E.2 Power comparisons of CM(Ref) - V against Exponential E-15

237 Power of CM(Ref)-V against EXP for n=5 Power of CM(Re)-V against EXP for n=10... i Significance Levels Significance Levels Power of CM(ReD)-V against EXP for n=15 Power of CM(Ref)-V against EXP for n=20 1! Significance Levels Significance Levels Figure E.2 (Continued) E-16

238 Power of CM(Ret-V against EXP for n=30 Power of CM(Re)-V against EXP for n= " " f Significance Levels Significance Levels Power of CM(Rel)-V against EXP for n=40 Power of CM(Ref)-V against EXP for n T g' Significance Levels Significance Levels Figure E.2 (Continued) E-17

239 Uall H!00eC0 s 0C04 I I CUS A e ab-0 0I *qo *q O. G~ O Cl ao 1.01 V S : 0 0 a S0IsIN 1 0 A 1,. - :0II Am..0 a w ~ f 000 " 0110 't 't0. C -. C b 0w 1 bc... 0G '000C N.00 C,1 100 a:00g 02 -O O G~~@ C@I 2... eo..o ~ ~ o b ~0 v0 0 " SO 'CIDE-18

240 0 0 'A' a Ptr- P0PI - v e b , a -e. a 6 *.r* a Fýr IP r i m1~e.io-0 q o~. 0 ~ W W ~ o.0o*0 ~ eq - eaqo 0.400~~C a0 "a IV,00000 e. 4 a- * * a eq no o.oo e o~~~~~ O O f *

241 m "" 1 e -4 e. li mee.m 4! 4 a...4 ms0e*.ew, 4! '3 It ; r ; -- rd.. 4! d! 4!d 10 We ;Zrw.. e 0e 3 a,* OWV1. a "0me awm~ ic, C a !,. ~ It m ~ o It.me fle 4! G! 4 d! - 01 am~ 01. m~.0 'a ~ m c; ei 'a"0e C m ' Ira.1!.ai- Z- -a a 2 M e e e e m A.ca.. m. o....~ C0 nm.... -e.. e!ee.4 Vo. r"i.. am.0 26m O a,.a miclib-e~a. eoc~aaf w.e - e,e. m P-Oo ;; a iv Is fe re omeme mm a.of;e~ c;.0.. aa-a-0 *0 e ml-e tan m 0ao- P Pa-- 1mmPi-a-a- *epem I a q wmi eov~ ~ m. a no.1e 00 lov. a a ow 2 a e I -a f ~ ~ ~ ~ ~ -. Ji c;f P ; r; ; IR1 4 E-0 ; a.0.00

242 al a as.. 00 "aa Ite -0 v Wn.'. a Ow a 0 *4 op Ve vg a 00:~' ae ao 0 ae.2aa wa i~~ I-0 10o a o me,~ Pe- P--a P- Ie~n'w reena a. we ne e-oaae- P P '. * *16.e. woe. flea mew. lie e. n n aa '*. e ee e em e m C, eenaneqa.eeae *.o t e.0a gwbwn. w C 0 ~ l nafmfta8~8eea~bgo a mo a.la e. Cwo wboeaý me. egwoa.ll'~ C -. ~~r emcee 14~e 1enmo 11 1; 1.. P- -P anpeer o oe..n woa.uwo s~e~a e~~~~nen~.. a e.. u- On m. C n.' 0.n - e ae..n e... m 4! 44!4!E! 14

243 0-0 o~~~d 4! ~ * 000 P;... ::.0 Me. or a m~: 0 0,*000 **F00 0* dt *. It0 Ca 0. a.a... 0 I. b a a 3! o, 00 c; P It".nC0 P- P 1 a r,-.a,,e P... e*2.47 o rn-ecee 11; n n-.s ee e n. g t O e I6.f 0b-b-. a.e V ; e 3!n. n 00.. n ae S ea ( d~o~oo oo e ooe0000=eoe1eoe~ I..E.

244 Power of CM(Ref)-V against BETA for n= I. I : pw of V *.. pw of CM(ref) I......: pe of Seq CM(reo-V Significance Levels o Power of CM(Ref)-V against BETA for n ~ :pw of CMjref) : * pe of Seq CM(reO-V 0 I.... I I Significance Levels Figure E.3 Power comparisons of CM(Ref) - V against Beta E-23

245 Power of CM(Ref)-V against BETA for n=5 Power of CM(Ref)-V against BETA for n= ,,., S L 1 0 z - I Significance Levels Significance Levels Power of CM(Ret)-V against BETA for n=15 Power of CM(Reo-V against BETA for n= bo i... i Significance Levels Significance Levels Figure E.3 (Continued) E-24

246 Power of CM(Ref)-V against BETA for n=30 Power of CM(Ret)-V against BETA for n= o... ~ ~ OA Significance Levels Significance Levels 1 Power of CM(Ref)-V against BETA for n=40 1i Power of CM(Reo-V against BETA for n= Significance Levels Significance Levels Figure E.3 (Continued) E-25

247 P-P Pn~ P. - I I!eaa-. I a ee r 1? "a n a- omee co 0. a* n*o. * ~ 1 ra P*~. r 0~nm.~~ A..1e- n.. 0 e 1;~ 1;.. P!a r; 1* -117! PP. P~nm P! r.-!i II.. -1;.. e..m w.- e.mw.e a. fm. WnI (i00, 23 " n.... *, 0 Z~ c; e. e : ~ -. In. I. I.. r... o 1-0. ~. -t r! - atella "r~~-- 9 n -V ai a he q ti.ak.c.cqvm1! wi-o c.'0ti.iv!.m m aaon eq ~ ~ ~ ~ ~ ~ ~~- rnakeaantaae lo 1'wma~ a;. e..1ee -. m e. ea q a q a a a a.e a.0 ;.a I Ie.. e 0ew.aaeN cc.aaaanva a--i-.wo. ;e Ine, a ew..m 00qqeee 10 a., '"a ii ea 2ra... oes nem O I ~ " ~. w Ieeq,..noA aeeekka,~. 0..aea 0.m Rmwwo a-31aa1.aeq o aaamq ee.m~am mqa~, a; aak-n miwwlk-! aa a a 12sm: a. - avmaa~e~a~waaaqv..: ea. aa...eeeqqqmm vm mmm m S aaaaaaooeaa aaaaaaaaooooama! :I~r P 2 4'a-n6

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