Confidence Intervals and Tests on the Difference of Means of Two Delta Distributions

Transcription

1 Western Michigan University ScholarWorks at WMU Dissertations Graduate College Confidence Intervals and Tests on the Difference of Means of Two Delta Distributions Karen Grace Villarente Rosales Western Michigan University Follow this and additional works at: Part of the Statistics and Probability Commons Recommended Citation Villarente Rosales, Karen Grace, "Confidence Intervals and Tests on the Difference of Means of Two Delta Distributions" (2010). Dissertations This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact

2 CONFIDENCE INTERVALS AND TESTS ON THE DIFFERENCE OF MEANS OF TWO DELTA DISTRIBUTIONS by Karen Grace Villarente Rosales A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Department of Statistics Advisor: Joshua D. Naranjo, Ph.D. Western Michigan University Kalamazoo, Michigan December 2010

3 CONFIDENCE INTERVALS AND TESTS ON THE DIFFERENCE OF MEANS OF TWO DELTA DISTRIBUTIONS Karen Grace Villarente Rosales, Ph.D. Western Michigan University, 2010 Various research fields produce data that are lognormally distributed and inflated with zero values. This type of data follows a delta distribution. In this study, we want to extensively investigate different interval and hypothesis testing methods for comparing the means of two delta populations to see which methods are optimal under different conditions of the populations. For confidence intervals, existing MVUE methods are extended to two sample cases and compared to classical and two proposed robust methods. We investigated the performance of the classical Student's t, Welch t, and Wilcoxon-based interval to see if these methods really perform badly on zero-inflated data. A simulation study is done to assess coverage accuracy of all the methods. Hypothesis testing on the equality of means of two delta distributions is done using confidence intervals. The previous interval methods are compared with two-part models by Lachenbruch (2001), MLEbased intervals, and two proposed robust intervals. The performance of the tests are assessed through a simulation study where the Type I error rates and power rates are computed. A robustness study is conducted by comparing the performance of estimators and tests under various distributions like gamma and Weibull.

4 Additionally, the Wilcoxon-based interval is assessed on three parameters that measure the difference of the two delta distributions namely, difference of means, difference of medians and median of differences. A simulation study is conducted to evaluate the performance of the three parameters of differences. The Wilcoxon-based interval was found to measure the median of differences best.

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7 2010 Karen Grace Villarente Rosales

8 ACKNOWLEDGMENTS My sincerest gratitude goes to my advisor, Dr. Joshua Naranjo. His guidance and encouragement helped in the completion of this project. I would also like to thank the members of my committee, Dr. Bugaj, Dr. Wang and Dr. Tena, for their time and expertise. To my husband, Mathew, thank you for your help and advices. Last but not least, thank you to my inspirations and sources of strength: Matt, Justin, Nathan and my parents, Dulce and Inigo. Karen Grace Villarente Rosales ii

9 TABLE OF CONTENTS ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES ii vi viii CHAPTER 1 INTRODUCTION Background and Motivation The A-Distribution Estimators of <5, k and v Review of Related Literature 6 2 INTERVAL ESTIMATION Classical Methods Pooled-t Confidence Interval Welch's t-confidence Interval Confidence Interval based on Wilcoxon Rank Sum Test Intervals Based on MVUE Aitchison and Pennington's Estimators Using Asymptotic Variance Estimate of k Proposed Confidence Interval 14 iii

10 Table of Contents - Continued CHAPTER Al-Khouli's Approach Proposed Confidence Intervals 19 3 HYPOTHESIS TESTS Lachenbruch's Two-Part Models MLE-based Methods Proposed Tests: Extension to MLE-based Intervals 25 4 WILCOXON-BASED CONFIDENCE INTERVAL Difference of Means Difference of Medians Median of Differences 29 5 SIMULATION STUDIES AND RESULTS Confidence Interval Results On A-distribution: Results and Discussion On Gamma and Weibull Distributions: Results and Discussion Hypothesis Testing Results and Discussion On A-distribution: Results and Discussion On Gamma and Weibull Distributions: Results and Discussion Wilcoxon-based Simulation and Results 74 iv

11 Table of Contents - Continued CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH 81 APPENDICES A. Confidence Intervals for A-distribution Case A 83 B. Confidence Intervals for A-distribution Case B 94 C. Confidence Intervals for A-distribution Case C 105 D. Confidence Intervals for A-distribution with 100% Contamination from Gamma Distribution 116 E. Confidence Intervals for A-distribution with 100% Contamination from Weibull Distribution 132 F. Type I Error and Power Rates for A-distribution 148 G. Type I Error and Power Rates for A-distribution with 100% Contamination from Gamma Distribution 155 H. Type I Error and Power Rates for A-distribution with 100% Contamination from Weibull Distribution 159 BIBLIOGRAPHY 163 v

12 LIST OF TABLES 4.1 Combination of Parameters Case A - Combination of Parameters for equal /i and a Case B - Combination of Parameters for equal 5 and a Case C - Combination of Parameters for equal 5 and fx % CI under Ai(0.2, 0.5, 1) and A 2 (0.2, 0.5, 1) % CI under Ai(0.2, 0.5, 1) and A 2 (0.4, 0.5, 1) % CI under Ai(0.2, 0.5, 1) and A 2 (0.6, 0.5, 1) % CI under A^O.2, 0, 1) and A 2 (0.2, 0, 1) % CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.5, 1) % CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.9, 1) % CI under A^O.2, 0.5, 0.15) and A 2 (0.2, 0.5, 0.15) % CI under A x (0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 1) % CI under Ai(0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 2) Power Rates of Tests for Equal /j and a Power Rates of Tests for Equal 6 and a Power Rates of Tests for Equal <5 and yu, Case 3: Results for varying 5 (equal // and a 2 ) Case 5: Results for varying S (equal n and a 2 ) Case 8: Results for varying /j, (equal S and a 2 ) Case 10: Results for varying n (equal 5 and a 2 ) Case 13: Results for varying a 2 (equal 5 and /x) 77 vi

13 List of Tables - Continued 5.21 Case 15: Results for varying a 2 (equal 5 and ji) % CI under Ai(0.1, 0.5, 1) and A 2 (0.5, 0.5, 1) % CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.9, 1) % CI under A^O.2, 0.5, 0.15) and A a (0.2, 0.5, 2) 80 vii

14 LIST OF FIGURES 5.1 Coverage Probabilities vs. (/tj /c 2 ) for Case A: 8 varies Coverage Probabilities vs. (m k 2 ) for Case B: [i varies Coverage Probabilities vs. k 2 ) for Case C: a 2 varies Coverage Probabilities vs. Width for Case A: 5 varies Coverage Probabilities vs. Width for Case B: /j, varies Coverage Probabilities vs. Width for Case C: a 2 varies Comparison of density plots for A(0.2, 0.5, 1) and A(0.2, 0.5, 1) Comparison of density plots for A(0.2, 0.5, 1) and A(0.4, 0.5, 1) Comparison of density plots for A(0.2, 0.5, 1) and A(0.6, 0.5, 1) Comparison of density plots for A(0.2, 0, 1) and A(0.2, 0, 1) Comparison of density plots for A(0.2, 0, 1) and A(0.2, 0.5, 1) Comparison of density plots for A(0.2, 0, 1) and A(0.2, 0.9, 1) Comparison of density plots for A(0.2, 0.5, 0.15) and A(0.2, 0.5, 0.15) Comparison of density plots for A(0.2, 0.5, 0.15) and A(0.2, 0.5, 1) Comparison of density plots for A(0.2, 0.5, 0.15) and A(0.2, 0.5, 2) Gamma Coverage Probabilities vs. Width for Case A: 5 varies Gamma Coverage Probabilities vs. Width for Case B: /i varies Gamma Coverage Probabilities vs. Width for Case C: a 2 varies Weibull Coverage Probabilities vs. Width for Case A: 8 varies Weibull Coverage Probabilities vs. Width for Case B: \i varies Weibull Coverage Probabilities vs. Width for Case C: a 2 varies 62 viii

15 List of Figures - Continued 5.22 Power curves for Case A: 5 varies (equal /i and a 2 ), at n = Power curves for Case B: /j, varies (equal 5 and a 2 ), at n = Power curves for Case C: a 2 varies (equal 8 and /i), at n = Power curves for Gamma Distribution, at n = Power curves for Weibull Distribution, at n = ix

16 Chapter 1 INTRODUCTION 1.1 Background and Motivation Various studies have been explored on data sets that are inherently positive and contain a large number of zeros. In some cases, the skewed values are distributed as lognormal. This type of distribution is referred to as delta distribution (A - distribution) a model first mentioned by Aitchison (1955) and described in Aitchison and Brown (1957). They proposed this two-part model to best handle data sets with zero and non-zero parts. Inferences on one sample mean and comparing two means are frequent points of interest in research studies on lognormal distributions. Zhou and Gao (1997) proposed confidence intervals for the one-sample lognormal mean. Zhou and Tu (1999) explored a likelihood ratio test when the populations contain both lognormal and zero observations. But methods for the difference of lognormal means have not been adequately explored (Chen and Zhou 2006), more so if the lognormal data is inflated with zero observations. Numerous methods are also available for estimating a single delta mean. Owen and DeRouen (1980), Pennington (1983), Zhou and Tu (2000), Fletcher (2008), 1

17 and Rosales (2009) investigated different confidence intervals for the mean of a delta distribution. Unfortunately, methods for the two sample situations are not well explored and understood. Until Zhou and Tu (2000), no confidence intervals have been proposed for methods that compare the means of two delta populations. Although approaches are available for these situations, detailed comparisons are lacking. Information about methods on the difference of two delta means are particularly hard to find. As a result, it is uncertain which of the methods are most appropriate. It is also unclear how the performance of these approaches might vary considering the sample sizes, the proportion of zeros, the population means, and the population variances. In this paper, both confidence intervals and tests on the difference of means of two delta distributions will be explored. We consider interval estimation techniques in our goal of not only testing the difference of the means but also providing information on the magnitude of the relative difference in the two population means. This study primarily aims to compare the means of two delta distributions through confidence interval and hypothesis testing methods and make recommendations on which methods are optimal under certain conditions. Specifically, this study will have the following objectives: Propose confidence intervals for estimating the difference between means of two delta distributions. Conduct a simulation study to compare the performance of existing and proposed confidence intervals based on their coverage probabilities. Propose tests for significant difference between means based on the confidence intervals previously proposed. A simulation study will be conducted 2

18 to compare the size and power of proposed and existing tests. Compare robustness of the confidence intervals and tests under various deviations from the delta distribution. Run and compare the preformance of the wilcoxon-based confidence interval on three parameters of interest difference of means, difference of medians and median of differences of two delta distributions. Chapter 2 will explore the different confidence interval methods for the difference of two means of delta distribution. Proposed, existing and classical methods will be discussed in detail. In Chapter 3 different hypothesis testing methods will be investigated based on all the confidence intervals discussed in Chapter 2. Additional existing 2-part tests by Lachenbruch (2001) will be compared to the methods discussed in the previous chapter. MLE-based methods will also be considered. There will be two proposed tests, through confidence intervals, that will be explored in this chapter. Power and size of the tests will be investigated. The Wilcoxon-based confidence intervals on difference of means, difference of medians and median of differences of two delta distributions will be explored in Chapter 4. Simulation studies and discussion of results will be presented in Chapter 5. Robustness of the methods will be investigated through simulation by designing two frameworks: using different combinations of parameter values and assuming that the positive part of the data comes from skewed distributions other than lognormal. 3

19 1.2 The A-Distribution The populations of interest are assumed to contain both zero and positive observations which follow a Delta distribution. It is denoted as A(5j,/Zj,erJ) with probability density function (1.1) The probability of having a zero response from the f h population is Sj, and that the positive values follow a lognormal distribution with mean fij and variance a? of the log scale, where 0 < Sj < 1 for j = 1, 2. If Sj = 0, the f h population has a lognormal distribution. If a random variable Yj of the f h then the mean and variance of the f h population has a A(Sj,pij,aj) distribution, population, which are functions of Sj, fij, and cr, are respectively given as E[Yj\ =(1 -Sj)e^V 2 uj= Var[Yj] = (1-5j)e 2^(e^ - (1 - S,)) (1.2) (1.3) Estimators of 5, k and v Let Y\j,...,Y nj.j be a random sample from the the f h population, that is, Y n, F 2 i,, ^nil be a random sample from the 1 st population, and Yi2,Y 2 2,,Y n2 2 be a random sample from the 2 nd population. We assume, without loss of generality, that in the f h sample the riji nonzero observations come first and the rest of the n 3 o = tij riji 4

20 observations are zero. For Yy > 0, log Y i3 jn^ ~ N(fij,a 2 ) for i = 1,...,riji] and Yij = 0, for % riji + 1,...,rij. Note that n^ ~ BIN(rij, 1 Sj). Under this model formulation the log-likelihood is ^(VS 2 r "ji n oo log Sj + riji log(l - Sj) - nj! log(%^\/27r) - j=l L i=i ~ H? J (1.4) where Xij = log y^ and ij) is the vector of parameters. Maximimizing the previous equation, we obtain the maximum likelihood estimators for Sj and fij to be, 5j = Tijo/nj H = n n v.j 1 x ij riji (1.5) (1.6) respectively, which are also unbiased. Since a 2 has a MLE of s 2 which is biased, a bias-corrected maximum likelihood estimator for a 2 will be used and is defined by, 5- SK tow X. ) 2 X (1.7) rij\ 1 riji 1 The mininum variance unbiased estimators of the mean and variance of the A- distribution were derived by Aitchison and Brown (1957) as functions of Sj, fij and S? They are given respectively as e^gn.. i rij "jl \ 2 Xjl if riji > 1 if riji = 1 if riji = 0 (1.8) J = 5

21 0 ^ (2s]) - ^G n j l ( ^ s 2 ) } if riji > 1 if %1 = l (1-9) if riji = 0 where G njl (t) is a Bessel function defined as, ( njl -1 )»- n\ x ( njl + l)( njl + 3) (nj! +2i - 3)i!' (1.10) kj is Pennington's (1983) expression for the unbiased estimator of the variance of &pen ( kj ) < 1.3 Review of Related Literature Aitchison (1955) first described a distribution that contains both zero and positive values in an application to household expenditures specifically on children's clothing. Some households spend nothing on children's clothing while others allocate high amounts that make the distribution highly skewed and frequently follow the lognormal curve. For interval estimation of the mean of delta distribution, Owen and DeRouen (1980) suggested to use the unbiased estimates of S, /i, and a 2 from the sample which are 6, and a 2, respectively, to construct a confidence interval. 6

22 On marine surveys, data are frequently inflated with zeros. Pennington (1983) looked into this example and examined a series of ichthyoplankton survey where it was aimed to estimate the total egg production of Atlantic mackerel in the study region. Also using Aitchison's mean and variance, he derived an estimate of the variance of the estimator of the mean to make a minimum variance unbiased estimator (MVUE) confidence interval. Zhou and Tu (2000) explored different methods in constructing confidence intervals for the mean of delta distribution namely, percentile-t bootstrap interval and two likelihood-based intervals. Fletcher (2008) investigated a profile-likelihood approach, which is also known as signed log-likelihood ratio approach, in constructing a 95% confidence interval for the mean of delta distribution. He cited an example on fisheries taken from a trawl survey where some trawls have no red cod while others have a positive density. They focused on estimating the mean density of red cod in the study area. To compare the means of two lognormal distributions, Zhou, Gao and Hui (1997) proposed two tests, the Z-score test and a nonparametric bootstrap approach. They illustrated these methods by comparing the medical costs of two groups of patients with type I diabetes. It was evident that the medical charges are very high for some patients resulting in significantly skewed distributions. In their simulation study comparing five different methods, the Z-score test was the best method. On the other hand, Wu et al. (2002) explored the same topic in a variety of small sample setting like small sample bioavailability study. They concentrated on getting p-values in testing the equality of means and constructing confidence 7

23 intervals for the ratio of means of two lognormal distributions and proposed two likelihood-based approaches the signed log-likelihood ratio and the modified signed log-likelihood ratio statistics. They said that the Z-score test does not perform well in a range of small-sample settings. Chen et al. (2006) investigated the ratio and difference of two lognormal means by comparing different interval estimate ideas on likelihood, bootstrap and generalized pivotal approach. They applied the techniques to examine the equality of means on medical cost data. Although Zhou and Tu (1999) compared the means of two populations to test whether they are the same or not, it did not offer information on the amounts of the relative differences in the two population means, as a confidence interval would do. Until Zhou and Tu (2000), no confidence intervals had been proposed for methods that compare the means of two delta distributions. They proposed a maximum likelihood-based method and a bootstrap method in constructing confidence intervals for the ratio in means of medical costs data that contained both lognormal and zero observations. Lachenbruch (2001) discussed a two-part model which arises when the distribution of observation has a continuous positive part and a zero inflated part. In 2002, he detailed an example in health care economics specific on hospitalization costs in a health insurance plan. In a year, most plan members will have no hospitalization costs while some will acquire extremely high hospitalization costs, accounting for the zero and positively-skewed part of the distribution, respectively. 8

24 Chapter 2 INTERVAL ESTIMATION Assume that we have two independent delta populations and the parameter of interest is the difference of means, k\ K2- Information about the relative difference of the population means is often of interest which can be measured through confidence intervals (CI). We say that the 100(1 - a)% confidence interval for K\ /c 2 is such that P [(«i -Ki)eCI] = l-a This chapter will discuss different methods for constructing such confidence intervals. Section 2.1 will cover classical methods like Pooled t, Welch t, and Wilcoxon-based confidence intervals. These methods will be assessed and compared with the existing and proposed methods to see how well or badly they will perform under different data situations. Existing MVUE methods on estimating the mean of delta distribution will be utilized and extended to two sample cases in Section 2.2. Proposed confidence intervals based on MVUE and robust estimators will be discussed in detail in Section

25 2.1 Classical Methods From a practical point of view, comparison of means often consider pairwise methods like Pooled-t and Welch t. But these methods ignore the clumping at zero as well as any other distributional attributes of the data that might make the estimation or test more sensitive. Another method is to use a nonparametric method like Wilcoxon rank sum test or Wilcoxon-based interval on the complete data set. Zero is sometimes omitted which makes the distribution of response different; but among those responding, the distribution is the same. Since this method also ignores some important features of the data, we will further investigate this method on which point of interest it measures best among difference of means, difference of medians and median of differences. This in depth investigation will be presented in Chapter Pooled-t Confidence Interval The t-statistic was introduced in 1908 by William Sealy Gosset. For a two sample problem the pooled t, also known as Student's t-confidence interval, assumes that the two distributions have equal variances. Thus, the interval uses the pooled standard deviation in its formula. Thel00(l - a)% confidence interval is given by (Yi -? 2 ) - t a/2>df sj±- + ±, (Y 1 - Y 2 ) + t a/24f S p f~ + 1 V n\ n 2 V n i n 2 Tlj where Yj = ^ Y t j is the overall sample mean, t a / 2t df is the upper quantile of the i=i t-distribution, rij is the overall sample size, and the degrees of freedom df = nj + n 2-2. S p is the pooled standard deviation known as 10

26 (m - 1 )sl + (n 2-1 )sl ni + n 2 2 We refer to this method as t for the rest of this dissertation Welch's t-confidence Interval A 100(1 - a)% confidence interval based on Welch's statistic is given as Unlike in Student's t-confidence interval, the two distributions are assumed to have unequal variances. Thus, the denominator is not based on a pooled variance estimate. The degrees of freedom u associated with this variance estimate is approximated using the Welch-Satterthwaite equation ' ni»2 ' ~ 04,4 This method will be denoted as t-w in this paper Confidence Interval based on Wilcoxon Rank Sum Test Let A be the location shift, also called as treatment effect. Form all possible differences between the first group and the second group. The estimator A is the median of these differences, and known as the Hodges-Lehmann (1963) estimator. For a (1-a) confidence coefficient, determine the upper a/2 percentile point w a / 2 of the null distribution of Wilcoxon. This percentile point can be obtained in R. Set 11

27 ni(2n 2 + ni + 1) = ~ HI- W a/2. The 100(1 - a)% confidence interval (Al, Ajy), is given by: X L = 0 {Ca \ A u = O inin2+l ~ Ca) (2.1) where O^,..., O^ are the ordered values of all possible differences, rij is the sample size of the } th group, and C a is an integer that approximates the ordered value of the lower confidence interval. For large samples, C a can be approximated by n _ nm 2 ^ ^a/2 nin 2 (ni + n 2 + 1) 1/2 12 This method is denoted as W for the rest of this dissertation. 2.2 Intervals Based on MVUE The MVUE-based intervals assume approximate normality for k and (k\ k 2 )- A 100(1 - a)% confidence interval for the difference, K\ k 2, is generally given by («i - k 2 ) ± z a / 2^v(kx) + 9(k 2 ) where z a / 2 is the upper quantile of order a/2 of the standard normal distribution. Note that this approach will not provide the asymmetry that one would expect in a confidence interval for the difference of means of skewed distributions. 12

28 2.2.1 Aitchison and Pennington's Estimators This method uses the normal approximation of MVUE of the mean k and Pennington's estimator of the variance of k. Using these estimators that are shown in expressions (1.8) and (1.11), the 100(1 - a)% confidence interval for (/tj k 2 ) is («1 - «2) ± Z ol /2\jVpen{.ki) + Upen(k 2 ) This method will be referred to as MVUE1 for the rest of this dissertation Using Asymptotic Variance Estimate of k Aitchison and Brown (1957) derived the approximate (asymptotic) variance of the mean estimator k for large n and 5 values considerably less than 1.0. This is expressed as Uqo (ftj) n-i (l-sj)(2a 2 + aj) (2.2) In practice, the z^(k) cannot be applied directly since S, // and a 2 are usually unknown. Consequently, Owen and DeRouen (1980) obtained an estimate of the variance by using unbiased estimates from the sample given in expressions (1.5) - (1.7) which are Sj, jjj and s 2, respectively. Thus, the asymptotic variance estimate of kj is given as j(«j) = n-i Sj(l-Sj) + ( 1 + (2.3) Therefore, the 100(1 - a)% confidence interval for (ki ac 2 ) with large sample approximation of the variance is 13

29 («1 - K 2 ) ± Z a /2\]Vaciki) + i>oo(k2) We refer to this method as MVUE2 for the rest of this dissertation. 2.3 Proposed Confidence Interval Two confidence intervals are proposed based on Al-Khouli's (1999) research on the robustness of the mean and variance estimators of A-distribution. These robust point estimators are expected to yield robust confidence intervals as well. Al- Khouli's approach will be extended to the two MVUE-based methods presented in Section 2.2. We first present Al-Khouli's idea on robust estimators in the succeeding subsection. The proposed robust confidence intervals based on MVUE will be explained in Subsection Al-Khouli's Approach Measures of location and scale of a distribution as well as their estimates are said to be robust if minor changes in a distribution have a relatively small effect on their values. A robust estimator is used mainly to prevent a small number of observation from having a significant effect on the estimate. Parameter estimation is generally just the first step in the analysis of data emerging from a linear model. Since test statistics are functions of these robust estimators they may result to robust tests and robust confidence intervals as well. Al-Khouli's research is essentially motivated by this idea - the invariance property of robustness. That is, a function of robust estimators is robust. 14

30 Since the sample mean and the sample variance both lack resistance and robustness, the potential influence of infrequent and very large observations is limitless. The estimators tend to overestimate and almost certainly, this overestimation will be inherited by the estimates of A-distribution. Additionally, the amount of overestimation can be very large, especially when exponentiated. In expressions (1.8) and (1.11), n and v are functions of 5, and a 2. The main objective is to find robust estimators for ji and a 2. Since there is no practical reason to robustify <5, the maximum likelihood estimator (MLE) 5 will be considered as an efficient estimator of 5. Based on these, Al-Khouli proposed to directly substitute x and s 2 with robust M-Estimators (Maximmum Likelihood Type Estimators) to obtain robust estimators of k and v. M-estimators are said to offer huge improvement in performance, flexibility and convenience. Different combinations of M-estimators for ji and a 2 were investigated by Al- Khouli. These M-estimators were then used to calculate the mean and variance of A-distribution. In particular, (Median, MAD), (T H, MAD) and (T H, S b ) were investigated where MAD is the median absolute deviaiton, is a one-step Huber M-estimator of locations and S& is a bi-weight A-estimator of scale. In a simulation study, Al-Khouli assessed the performance of each of these estiamtors and found that the pair (Th, Sb) was significantly robust when contaminants are present and had the most precise results in terms of relative efficiency and relative bias for small samples. In this paper, only the combination of one-step Huber M-estimator of location Th and bi-weight A-estimator of scale Sb will be considered. Huber M-Estimator of Location In general, M-estimators minimize functions of the deviations of each observa-

31 tion from the location estimate. Some special cases of class of M-estimators include the mean and the median. M-estimators generalize the concept of maximum likelihood estimator of a location parameter. It is reasonable to expect that a suitably chosen M-estimator will have good robustness and efficiency in large samples and will perform well in small to moderate sample size (Al-Khouli, 1999) Let T n. be the M-estimator of location then T nj is the value that will minimize where p is an arbitrary function (commonly known as the objective function). Or equivalently, T nj is the value that satisfies where ip(y, t) is the first derivative of p. Mostly, an M-estimator of location must take into account a scale in order to be location and scale equivariant. We define where S nj is an auxiliary estimator of scale and c is some tuning constant. But in using S nj, we need to simultaneously estimate T nj and S nj, which can be a complicated problem. However, Huber (1981) said that this extra complication is not necessary and suggested a simplified version by choosing a preliminary estimate of scale, that is, a fixed auxiliary estimate of scale So, Thus, M-estimator is the value that satisfies. (2.6) 16

32 Sometimes we cannot find an explicit solution for expression (2.4). A good approach is to solve this iteratively using the Newton-Raphson algorithm. Let Im) Vij ~ l/:. ' - and E n j i r (. m )l Ji nj Hi ( 2-5 ) ~ n i + where ijj' is the second derivative of the objective function p. The M-estimate of location is the limit of T^. For Huber M-estimator, the objective function is defined as U V 2 ou(u )-{ iov\ Uj \<d Ph\ u jj - _ k2j 2 for \ Uj \ > d. Consequently, and ( \ i u j f r l^jl < d tph^j)- <j k s g n^ f or > d 1 for \uj\ < d ^/K) - i o f or \ua>d Therefore, a fully-iterated Huber M-estimator can now be achieved by using (2.5). Initial estimate T^ and So j are set to be the sample median and median absolute deviation (MAD), respectively. When using Huber estimate, it is proposed to use the rescaled or normalized MAD (NMAD). NMAD is equal to 1.48 x MAD which only means that the tuning constant c is equal to An easier method is the so-called one-step M-estimator. From expression (2.4), the one-step Huber M-estimator is defined to be. (2.6) 17

33 The fully iterated and the one-step M-estimators are said to be aprroximately equally robust (Andrews, et al., 1972). To calculate the Huber M-estimators in R, the function hubers from the package MASS can be used. Bi-Weight A-Estimator of Scale Robust estimators of scale are used as auxiliary estimators in M-estimator of location. But they can be points of interest on their own. As a measure of variability, Shoemaker and Hettmansperger (1982) utilized the square root of the midvariance the asymptotic variance of a robust M-estimator. In 1982, David Lax first introduced the term A-estimator of scale. This refers to a finite sample approximation of the variance of M-estimator. Letting yij ~ Mj cmadj Note that s^ is called the sample midvariance (Shoemaker and Hettmansperger, 1982) of the j th population. For biweight estimators, we consider the functions ipbj and ip' b. defined respectively as for ttj < 1 for u 3 1 > 1 (2.7) for < 1 for \uj\ > 1 (2.8) 18 for \uj\ < 1 for \uj\ > 1. (2.6)

34 From expressions (2.7) - (2.9), the corresponding biweight ^-estimator of the j th population is given by (2.10) Biweight ^-estimator is better in performance than Huber A-estimator especially in the presence of extreme contamination. For biweight estimators, the reasonable value of the tuning constant is 6 < c < 9 (Al-Khouli, 1999) Proposed Confidence Intervals We propose robust equivalents of MVUE1 and MVUE2. These proposed methods are extensions of Al-Khouli's idea of robustifying intervals or tests by replacing nonrobust estimators, like /} and s 2, with more robust ones like and Sb, respectively. 1) Robustified MVUE1 For Aitchison and Pennington's MVUE of the mean and variance, let km, = nj 0 if riji = 1 if riji = 0 (2.11) and \nj> 0 if n,ji = 1 (2-12) if n.ji ~ 0 19

35 be the robustified mean and variance, respectively, where Tn j is the j th Huber M-estimator of location and Sj,. is the j th biweight A-estimator of scale. Then, the corresponding 100(1 - a)% robust confidence interval for MVUE1 is («Afi - «JW 2 ) ± + VM(km 2 ) This method is referred as R-MVUE1. 2) Robustified MVUE2 With Aitchison's MVUE of the mean (1.8) and Owen and DeRouen's MVUE of the varaince (2.3) of A-distribution, we let km. J n s 31 " 2 and 2T H.+S b. e 3 i ^ (i - Sj) (1-^(2S 6,+S 6 2?.) be the corresponding robust estimators of the mean and variance, respectively where Tn j is the j th Huber M-estimator of location and Sb 3 is the j th biweight A-estimator of scale. Thus, the 100(1 - a)% robust confidence interval based on MVUE2 is (k Ml - KM 2 ) ± z a/2 \/0 oo (km 1 ) + boo(km 3 )- We denote this method as R-MVUE2 for the rest of this dissertation. 20

36 Chapter 3 HYPOTHESIS TESTS The confidence interval methods discussed in the previous chapter will be utilized to conduct hypothesis tests on the difference of the two delta means. Specifically, we test H 0 : Ki - k 2 = 0 vs. Hi: K\ - k 2^ 0, or, H 0 : log(«i) - log(/t 2 ) = 0 vs. Hi: log(«i) - log(k 2 )^ 0. The decision is to reject the null hypothesis if the 95% confidence interval does not contain zero. Additionally, MLE based intervals will be carried out to perform the test. The performance of the different existing and proposed methods will be compared through type I error rate control and power. 3.1 Lachenbruch's Two-Part Models Additional objective of this study is to compare all the hypothesis testing methods discussed with the concept of two-part models by Lachenbruch (2001). The twopart models occur when a distribution has continuous positive responses and a clump of zero observations. Two tests from this idea will be evaluated, the two- 21

37 part t-test and two-part rank sum test. In the two-sample case, we are interested with #o:/i(y) = / 2 (y), or equivalently H o- My) = My) 9 iven y > 0 HS: 5! = where hi and h 2 are probability distribution functions of the nonzero part. H 0 is true if both Hq and Hq are true, and vise versa. First, evaluate the test of equality of proportion using the standard tests of binomial proportions. Then a t- test or Wilcoxon rank sum test is computed for the positive observations. Finally, compare the sum of the squares of the normal deviates corresponding to these tests to a x 2 distribution with 2 degrees of freedom. 1) 2-Part t-test The binomial test B for the equality of proportions of zeros is B = Ji - S 2 y/siil - 5^/rii {1-5 2 )/n 2 and t = The 2-part t-test, which we denote as T-P t, is given as X 2 = B 2 + t 2 22

38 This is distributed as y 2 with 2 degrees of freedom. 2) 2-Part Rank Sum Test The test statistic is X 2 = B 2 + W 2 where W is the Wilcoxon rank sum test statistic given by: where the sum is taken over the rank of observations in the first group. With E(W) = ni ( i)(ni(i)+ ni( 2 )+l)/2 and var(w) = n 1(1 )ni (2 )(n 1 ( 1 )+ n 1(2) +l)/12, the statistic is asymptotically normal, so W 2 is asymptotically x 2 with 1 degree of freedom. Therefore, the 2-part rank sum statistic is also distributed as x 2 with 2 degrees of freedom. We denote this as T-P w. For the two models, we reject Hq if the test statistic > % 2 = MLE-based Methods Intervals based on MLE assumed normality approximation of the MLE 6. Here, 0 = log (k). In general, the 100(1 - a)% confidence interval for the difference of (log K\ - log k 2 ) is (0 X - 2 ) ± z % y/uidi) + v(6 2 ). 23

39 1) Bias-Corrected Approach Through maximizing the log-likelihood of A-distribution in (1.3), 5, jl, and s 2 in expressions (1.4) - (1.6) are obtained. These are the maximum likelihood estimates of //, and a 2, respectively. Zhou and Tu (2000) proposed s 2 as an estimator of a 2 to remove the bias. Therefore, <?, can be estimated as l g(l Sj) + Pj + (3.1) Using delta method, _2 4 hc(0j) = + ^ + (3.2) rijuji riji zriji Through asymptotic normality of d, the corresponding 100(1 - a)% confidence interval for the difference (log - log k 2 ) with bias correction is: (0 X - 2 ) ± z a/2 \f u b c0i) + hc0 2 ) This method is denoted as MLE-BC. In as simulation study performed by Zhou and Tu (2000) for one-sample case, MLE-BC works best when skewness is small and performs satisfactorily for moderate to large sample sizes. It tends to have poor coverage for small sample sizes. 2) Truncated Binomial Approach Fletcher(2008) proposed a confidence interval using Aitchison's estimator of k which was an MVUE. He first constructed a confidence interval for 6 = log (k), an idea inspired by David Cox. Then he back-trasformed the endpoints. The estimate for 9j is given in (3.1). 24

40 Pennington (1983) provided a formula for the MVUE of the variance of the MVUE k. But Fletcher (2008) used a truncated Binomial approach restricting only to situations where the number of positive observations, n^i > 1. He came up with an estimate of the variance K " nji(l -cd)' n 2(n fl + l) 1 ' where c = (njo/n,j) n i 1 and d 1 + (rij l)(riji/nj) Due to the restriction in riji, the MLE of 8 is not 8 anymore and the "true" MLE can be found through numerical search (Finney, 1949). But the difference between the "true" MLE and <5 is very small that expression (3.4) is still valid to use (Fletcher, 2008). The 100(1 - a)% confidence interval for the difference (log k\ - log /c 2 ) is (0i - e 2 ) ± Z a / + This method is referred as MLE-TB. In one-sample case, for moderate to large sample sizes, this method tends to perform good enough (Fletcher, 2008). It gives poor coverage for small sample sizes and very high upper confidence limit when sample size is small and level of skewness is low. 3.3 Proposed Tests: Extension to MLE-based Intervals Since hypothesis testing will be done using confidence intervals, two intervals will be proposed based on Al-Khouli's (1999) research on the robustness of the mean 25

41 and variance estimators of delta distribution. These intervals will compare the log of means of two delta distributions. Based on the idea that robust estimators may lead to robust tests and robust confidence intervals, we proposed to expand the MLE-based confidence interval to a more robust form. This is done by substituting jl and s 2 in expressions (3.1)-(3.3) with robust etimators Th and Su, respectively. Thus we have M. = log{\ - & 3 ) + T Hj + SU2, (3.4) VM bc (0Mi) s b n j0 s^ rijtiji riji 2riji (3.5) and r/ \ _ {d - c)(l - cd) - njl(l - c) 2 s bj s b n jx (l - cd) 2 riji 2{n j i + 1) The corresponding 100(1 - a)% confidence interval for the difference (log k,\ log k 2 ) based on MLE-BC with robust estimators is (8Mi ~ 9m 2 ) ± z a/2 \]hc M 0M! ) + hc M ( M 2 ) This method is referred as R-BC. In the same manner, the corresponding 100(1 - a)% confidence interval for the difference (log K\ - log k 2 ) based on MLE-TB with robust estimators is (l9ml - Qm 2 ) ± z a/2\!&tbm 0Mi) + ^tb M 0M 2 ) We denote this method as R-TB. 26

42 Chapter 4 WILCOXON-BASED CONFIDENCE INTERVAL The Wilcoxon-based confidence interval is a nonparametric method that compares the difference of two distributions. This difference, which we denoted as A is a shift factor also known as the Hodges-Lehmann parameter. Let population 1 have a distribution function F(t) and population 2 have a distribution function G(t). A location-shift model described by Hollander and Wolfe (1999) is expressed as G(t) = F(t - A), for every t. (4.1) This chapter will discuss three different measurements that describe the location shift factor, A. Moreover, we want to find out how the wilcoxon-based confidence interval performs on these three parameters of interest namely, difference of means, difference of medians, and median of differences. The next three sections will provide theoretical derivations based on the assumption that F and G came from A-distribution. 27

43 4.1 Difference of Means We already established how to construct confidence intervals for ki k 2 in Chapter 2 and parts of Chapter 3 on different confidence interval methods. To assess their coverage probabilities and error rates, these methods were compared with the true mean difference given by k\ K2 = (1 - S^e^"' 2 - (1-52)e^+a^2 (4.2) More often than not, applications in medical and biological fields often opt for difference of means, as a method to compare two distributions, to take into account every observation into the analysis even if the data are higly skewed. 4.2 Difference of Medians When distributions are highly skewed, like Lognormal, it is better to compare the difference of medians to withstand the possibility of overestimation due to extreme outlying values. If a random variable Yj of the f h population has a A(5j,/ij,a 2 ) distribution, then the median of the A-distribution denoted as m, is such that Fy (rn) = < m] = 0.5. With expression (1.1) as the probability density function of A-distribution, let H(Y) be the cumulative density function of the lognormal distribution. We have p[y < m] = <5 + (1 - S)H Y (m) (4.3) where //y (m) is given by 28

44 H Y (m) = ^ + ^er/ where erf is an error function defined as log TTI, [1 ry/2 (4.4) 2 f X 2 erf(x) = p: / e _t w i/tt 7O dt. (4.5) Equating (4.3) to 0.5 and solving for the median m will give us m = exp + o\p2,erf~ x 5-1 (4.6) Thus, the difference of medians of A-distributions is given by mi m 2 = exp pi + ai\flerf 1 ^ ^ - exp p 2 + cr 2 V2erf 1 (4.7) 4.3 Median of Differences Mann-Whitney-Wilcoxon is one of the competing estimators that we study. In comparing its coverage probability against other estimators, we should consider that the confidence interval is designed to estimate the median of differences, Med[Yi Y 2 ], rather than the difference in means, K\ k 2, or the difference in medians, Med[Yi] Med[Y 2 ]. This measure is essentially the parameter for the Hodges-Lehmann estimator. The median of the differences m is such that P[Yi - Y 2 <m]=8 1 + (l-& x )H Yl {m) (1-5 2 )H Y2 (m) = 0.5. (4.8) 29

45 The cumulative distribution function of (Yi Y 2 ) is expressed as F(Y 1 - Y 2 ) = <$i + (1 - «5 2 )(1 - Hy 2 (-t)) + ( )HYl-Y 2 (t) t < 0 Si + s 2 ( 1 - S^Hy^t) + (1 - <5i)(l - S 2 )H Yl _Y 2 (t) t > 0 (4.9) where H Y2 (-t) = I + W i + W I 12^] (4.10) (4.11) and H y [ _y 2 (i) is the cdf of the difference of two lognormal distributions which we derive subsequently. At this point, we restrict y to be positive. The probability distribution function, h(yj), of the j th lognormal distribution is given by KVi) = : exp flog Vi ~ N? 2a] (4.12) Our goal, at this point, is to derive the marginal pdf of (Yi Y 2 ) by first obtaining the joint density J(yi, 1)2) Let z = yi y 2 and a companion transformation, w = y\. An inverse transformation is given by yi= w y 2 = w- z. (4.13) where we can further let 7(z, w) = w and 77(2, w) = w z. The Jacobian of the transformation is the determinant of J = d'y d'y dz Qw 0 1 di) or) dz dw

46 which is J\ 1. Thus, the joint density of (yi,y 2 ) is f(yi,v2) = 2nyiy2 a 1 a 2 exp -(log yi - pi) 2 (log y 2 - // 2 ) 2 2a? 2a 2 (4.14) So the joint density of (Z, W) is fw,z(w,w - z) = 2ttw(w z)o\o2 exp -(logw-/ii) 2 (log(w - z) -/J, 2 f 2 al 2 a 2 (4.15) Then we compute the marginal pdf gz(z) of z = y\ y 2 as follows 9z{z) roo / f(w, z)dw z < 0 f(w, z) dw z > 0 (4.16) Now, the cdf of g(z), which is //^^^(Z,) in expression (4.9) is H Yl -y 2 (t)= f gz(z)dz (4.17) J 00 Now that we have defined all the terms of the cdf F(Y\ Y 2 ) we can now compute for the median of the differences m in expression (4.8). The software R will be used to code and compute for the value of the median of differences. Some parameter values of the A-distribution are used to solve for and compare the three measurements previously discussed. Table (4.1) shows different values for the difference of the two A-distributions through difference of means («i k 2 ), difference of medians (mi m 2 ), and median of differences m. 31

47 Table 4.1: Combination of Parameters Case Si s 2 W ^ mi 7712 m

48 Chapter 5 SIMULATION STUDIES AND RESULTS To assess the general performance and robustness of the interval estimators simulations on the A-distribution are considered in Section 5.1 under different combination of parameters. For further test on robustness, simulation will also deal with misclassification of the data as coming from other skewed distributions similar to lognormal, like gamma and Weibull. In Section 5.2, we evaluate the Type 1 error rates and power rates of the tests considering that the data come from A-distribution. Then we run simulation for data where the positive parts come from gamma and Weibull distributions with the same mean and variance as the lognormal distribution. The final section will cover the simulation framework and results of the Wilcoxon-based confidence interval on difference of means, difference of medians and median of differences. Robustness of the different confidence interval methods will be assessed using the following criteria: Coverage Probability (CP): proportion of intervals such that the true parameter falls within the interval 33

49 Coverage Error (CE): absolute difference between coverage probability and nominal probability Lower Error Rate (LER): proportion of intervals such that the true parameter falls below the interval Upper Error Rate (UER): proportion of intervals such that the true parameter falls above the interval Average Width (Width): average of the widths of simulated intervals In the course of the simulation study, we set the nominal probability to 0.95 and nominal significance level to Thus, we prefer to have CP=0.95, CE=0, LER=0.025, and UER= Also, wa want the average width to be as small as possible. For all the simulations, each case employs equal sample sizes of 15, 25, and 50 for both groups. Ten thousand simulations are done for each case at a specific sample size. 5.1 Confidence Interval Results This simulation methodology considers cases where one parameter is changing. Table 5.1 shows the combination of parameters where only the proportions of zeros 5 are changing. The value of the means are equal and set to 0.5, while both the variances are set to 1.0. The parameter combination where only the means are unequal is shown in Table 5.2. The proportion of zeros are equal for both groups and set to 0.1, 0.2, and 0.4, while the variances are fixed to 1.0. Lastly, the parameter combination where the variances are unequal is presented in Table 5.3. The means are equal and set to 0.5, and the proportions of zeros are also equal where the values 0.1, 0.2, and 0.4 are used for simulation. 34

50 Table 5.1: Case A - Combination of Parameters for equal /j, and Case Ji a 2 <5i fa al a2 0.2 a3 0.3 a4 0.4 a5 0.5 a a7 0.3 a8 0.4 a9 0.5 alo 0.6 all al2 0.4 al3 0.5 al4 0.6 al5 0.7 Table 5.2: Case B - Combination of Parameters for equal 5 and Case 5 a 2 Ml M2 bl b2 0.3 b3 0.5 b4 0.7 b5 0.9 b b7 0.3 b8 0.5 b9 0.7 blo 0.9 bll bl2 0.3 bl3 0.5 bl4 0.7 bl

51 Table 5.3: Case C - Combination of Parameters for equal 5 and fi Case,5 M A cl c2 0.5 c3 1.0 c4 1.5 c5 2.0 c c7 0.5 c8 1.0 c9 1.5 clo 2.0 ell cl2 0.5 cl3 1.0 cl4 1.5 c!

52 5.1.1 On A-distribution: Results and Discussion The difference of means, are assessed and compared using the following estimators: t - two-sample t (see p. 10) tw - Welch's t (see p. 11) W - Wilcoxon (see p. 11) MVUE1 - Pennington, 1983 (see p. 13) MVUE2 - Owen and DeRouen, 1980 (see p. 14) R-MVUE1 - Robustified Pennington (Al-Khouli, 1999) (see p.19) R-MVUE2 - Robustified Owen and De Rouen (see p. 20) Only selected cases are presented here. The rest of the tables are shown in Appendix A - C. In general, both t and tw perform very well under the null cases. Although W has the shortest width for most cases, it does not maintain good coverage as the difference, K\ increases and as the sample sizes increase. This is evident in Figures where coverage probabilities of Wilcoxon are already low for moderate differences in means but drops even more for large differences. Also Figures show that Wilcoxon has the narrowest width among all the methods but its coverage probability is far from nominal level of With this in mind, we can infer that Wilcoxon-based interval does not perform very well on the difference of means of two delta distributions. But since Wilcoxon-based interval is known to perform well under skewed type of distributions, we will have 37

53 an in depth investigation on this method in Section 5.3 as to which parameter of difference it measures well. Figures 5.1 and 5.2 also show that varying values of <5 and // have little effect on the coverage probabilites of all the methods (except W). But when a 2 varies, as shown in Figure 5.3, the coverage for most of the methods decreases significantly for any values of n. Figures also show that MVUE2 has the widest interval for all the cases. For cases where the means and variances are equal and the proportions of zeros are unequal (Case A), results are presented in Tables The comparison of the density plots are shown in Figures As the difference in 5 for both groups increases, R-MVUE1 tends to perform very well in terms of having a short width and good coverage. For the rest of the cases, MVUE1 and the ts also have good coverage and relatively short width. We can say, specifically, that the ts' performances are not affected if the data contain small to large proportions of zeros. Tables present results where only the means varied (Case B), while corresponding comparison of density plots are shown in Figures R- MVUE1 performs best with the shortest width and relatively good coverage; while MVUE1 has very similar performance. MVUE2 has a coverage probabilty closest to nominal for all small to large sample sizes when the difference in means are large. In the last cases where only the variances are unequal (Case C), results are presented in Tables and density plots are shown in Figures For small to moderate differences in the variances, ts and MVUEs have good coverage, especially as n increases. But all the methods lack coverage as

54 the difference in the variances become large. Among the methods, MVUE2 has coverage probabilities closest to nominal level but this is compensated by having the widest interval. n=15 n=25 n=50 mean difference - t tw mean difference mean difference x MVUE1 - >- MVUE2 V R-MVUE1-H- R-MVUE2 Figure 5.1: Coverage Probabilities vs. (ki k 2 ) for Case A: 5 varies 39

55 n=15 n=25 n= r mean difference t -A- tw + W mean difference x MVUE1 -O- MVUE2 -v R-MVUE1 - mean difference - R-MVUE2 Figure 5.2: Coverage Probabilities vs. (ki «2 ) for Case B: // varies 40

56 n=15 n=25 n= r T i 1 1 r mean difference mean difference mean difference tw + W x - MVUE1 -O- MVUE2 -v R-MVUE1 -a- R-MVUE2 Figure 5.3: Coverage Probabilities vs. (/ti k 2 ) for Case C: a 2 varies 41

57 Design a8, n=15 Design a10, n=15 0. o OJ O _ CO <r width width Design a8, n=50 Design a10, n=50 CL O width width Legend: 1 - t, 2 - tw, 3 - W, 4-MVUE1, 5-MVUE2, 6-R-MVUE1, 7 - R-MVUE2 Figure 5.4: Coverage Probabilities vs. Width for Case A: 5 varies 42

58 Design b8, n=15 Design b10, n=15 Q. o width width Design b8, n=50 Design b10, n=50 o. o i i i i r width o width Legend: 1-t, 2-tw, 3-W, 4-MVUE1, 5 - MVUE2, 6 - R-MVUE1, 7 - R-MVUE2 Figure 5.5: Coverage Probabilities vs. Width for Case B: ^ varies 43

59 Design c8, n=15 Design c10, n=15 a. o o> o I- o Q. O r- o width width Design c8, n=50 Design c10, n=50 Q. o oo o CO o a. O c> CM O O O width width Legend: 1 - t, 2-tw, 3 - W, 4-MVUE1, 5 - MVUE2, 6 - R-MVUE1, 7 - R-MVUE2 Figure 5.6: Coverage Probabilities vs. Width for Case C: a 2 varies 44

60 Table 5.4: 95% CI under Ai(0.2, 0.5, 1) and A 2 (0.2, 0.5, 1) k x = , K 2 = , K x - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUE R-MVUE Delta(0.2,0.5,1) Del(a(0.2,0.5,1) S 6 Figure 5.7: Comparison of density plots for Ai(0.2,0.5,l) and A 2 ( ,l) 45

61 Table 5.5: 95% CI under A x (0.2, 0.5, 1) and A 2 (0.4, 0.5, 1) ki = , k 2 = , Kl - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUE R-MVUE Delta(0.2,0.5,1) Detta(0.4,0.5,1) n Figure 5.7: Comparison of density plots for Ai(0.2,0.5,l) and A 2 ( ,l) 46

62 Table 5.6: 95% CI under Ai(0.2, 0.5, 1) and A 2 (0.6, 0.5, 1) ki = , k 2 = , - k 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE Delta(0.2,0.5,1) --- De!ta(0.6,0.5,1) Figure 5.7: Comparison of density plots for Ai(0.2,0.5,l) and A 2 ( ,l) 47

63 Table 5.7: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0, 1) K\ = , K 2 = , - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUE R-MVUE t 50 ' tw W MVUE MVUE R-MVUE R-MVUE Delta(0.2,0,1) Delta(0.2,0,1) ~~i r Figure 5.7: Comparison of density plots for Ai(0.2,0.5,l) and A 2 ( ,l) 48

64 Table 5.8: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.5, 1) ki = , K2 = , - «2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE Detta(0.2,0,1) Delta(0.2,0.5,1) ~ Figure 5.7: Comparison of density plots for Ai(0.2,0.5,l) and A 2 ( ,l) 49

65 Table 5.9: 95% CI under A x (0.2, 0, 1) and A 2 (0.2, 0.9, 1) ki = , k 2 = , - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUE R-MVUE Detta(0.2,0,1) Delta(0.2,0.9,1) d I x Figure 5.14: Comparison of density plots for Ai(0.2,0.5,0.15) and A 2 (0.2,0.5,1) 50

66 Table 5.10: 95% CI under Ax(0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 0.15) K! = , K 2 = , - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE Figure 5.13: Comparison of density plots for A x (0.2,0.5,0.15) and A 2 (0.2,0.5,0.15) 51

67 Table 5.11: 95% CI under Ai(0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 1) «i = , «2 = , m - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUE R-MVUE Figure 5.14: Comparison of density plots for Ai(0.2,0.5,0.15) and A 2 (0.2,0.5,1) 52

68 Table 5.12: 95% CI under A x (0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 2) «i = , k 2 = , m - k 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE Figure 5.15: Comparison of density plots for Ai(0.2,0.5,0.15) and A 2 (0.2,0.5,2) 53

69 5.1.2 On Gamma and Weibull Distributions: Results and Discussion For cases where the positive part of the data comes from gamma or Weibull distribution, simulation is done using cases a6 - alo (Table 5.1), cases b6 - blo (Table 5.2), and cases c6 - clo (Table 5.3). Results are shown in Appendix D for gamma distribution and Appendix E for Weibull distribution. For the gamma distribution where only the <5s are unequal, the ts are best overall. This is seen in Figure Looking at Figure 5.17 for Case B where only the /i varies, ts and R-MVUEs perform well while all methods break down when the difference in /i becomes large. For cases where er 2 s are unequal, the ts perform satisfactorily as seen in Figure But all methods do not perform well when there is a large difference in cr 2 s between the two groups. To have more meanigful assessments on the graphs, MVUE2 (legend = 5) was ommitted in Figures since its widths are very large that details on the other methods are impaired. The detailed results of the Gamma simulation are as follows: CASE A-5 varies: For the null case, ts are best with a narrow interval and very good coverage. For all the cases, the ts perform best followed by R-MVUE1. The MVUEs break down with very wide intervals. CASE B - /i varies: For the null case, W is best while R-MVUE1 performs well for small n. The ts are very satisfactory as well. 54

70 For moderate differnce in /i between the two groups, the ts perform very well followed by R-MVUE1. o For large difference in R-MVUE2 is the best overall, but ts and R-MVUE1 perform good as well. All other methods start to break down either with low coverage and/or very wide intervals. CASE C-a 2 varies For the null case, ts and MVUEs are best. o For moderate differnce in a 2 between the two groups, the ts perform very well as n increases. o For large difference in a 2, all methods start to break down either with low coverage and/or very wide intervals. 55

71 Design a8, n=15 Design a10, n=15 ai o i a. o 10 ao o r-- o T 25 width width Design a8, n=50 Design a10, n=50 Q. o CO o D. o cq c> -ao width width Legend: 1 - t, 2 - tw, 3 - W, 4 - MVUE1, 6 - R-MVUE1, 7 - R-MVUE2 Figure 5.16: Gamma Coverage Probabilities vs. Width for Case A: 5 varies 56

72 Q. o 0.95 i i Design b8, n=15 Design b10, n= in 4 1 d r S in - 6 d o _ ID I-- - m - o 3 o ID CD - T ~r 40 in CD - 3 T width width Design b8, n=50 Design b10, n=50 CO o 0. CO O d Q. o CD d d CM d d CM d width width Legend: 1 - t, 2 - tw, 3-W, 4-MVUE1, 6 - R-MVUE1, 7 - R-MVUE2 Figure 5.17: Gamma Coverage Probabilities vs. Width for Case B: /i varies 57

73 Design c8, n=15 Design c10, n=15 Q. o CD d Q. O CM CD width width Design c8, n=50 Design c10, n=50 00 o oo d Q. O CO d o D. o CO d d CM d CM d width width Legend: 1 - t, 2 - tw, 3 - W, 4 - MVUE1, 6 - R-MVUE1, 7 - R-MVUE2 Figure 5.18: Gamma Coverage Probabilities vs. Width for Case C: a 2 varies 58

74 For the positive observations coming from Weibull distribution, Figure 5.19 shows the results where only the 8 varies. We can see that t is the best method for this case. For cases where only the ji varies, t is still the best method as shown in Figure For Case C where only the o 2 varies, Figure 5.21 shows that t is good for small to moderate a 2 difference. Due to extremely large width, MVUE2 are not shown in Figures 5.19 and Here are the overall results of the simulation for Weibull: CASE A - 8 varies: For the null case, W is best since it has the narrowest interval with coverage closest to the nominal level. The ts perform best as n increases. For all the cases where the <5 varies, the ts perform best followed by the R-MVUEs. R-MVUE1 is good for large differences in 5 between the two groups. CASE B - fj, varies: For the null case, the ts perform best with a narrow interval and good coverage. For moderate difference in ji between the two groups, the ts have the best coverage and narrowest interval as n increases. For large differences in 8 between the two groups, R-MVUE2 performs well for small n, while R-MVUE1 is good for large n. CASE C-a 2 varies 59

75 For the null case, the ts perform very well. For small to moderate difference in cr 2 between the two groups, the ts are still the best while RMVUE2 perform satisfactorily. For large difference in a 2 between the two groups, all the methods start to break down by having very wide intervals or low coverage. Design a8, n=15 Design a10, n=15 Q_ a width width Design a8, n=50 Design a10, n=50 CD d Q_ o <o c> in d width width Legend: 1-t, 2 - tw, 3 - W, 4-MVUE1, 6-R-MVUE1, 7 - R-MVUE2 Figure 5.19: Weibull Coverage Probabilities vs. Width for Case A: S varies 60

76 Design b8, n=15 Design b10, n=15 CL O r CL O in O) - d co in _ d to d CO _ width width Design b8, n=50 Design b10, n=50 oo d Q_ O CO d o Q. O co d -a; d CM d width width Legend: 1 - t, 2 - tw, 3 - W, 4 - MVUE1, 5 - MVUE2, 6 - R-MVUE1, 7 - R-MVUE2 Figure 5.20: Weibull Coverage Probabilities vs. Width for Case B: /i varies 61

77 Design c8, n=15 Design c10, n=15 CO ci CO d Q- O CO d 0. O CO d d d CM d width width Design c8, n=50 Design c10, n=50 CO d 0- o Q. o CO d f d CM d o width width Legend: 1 - t, 2 - tw, 3 - W, 4 - MVUE1, 6 - R-MVUE1, 7 - R-MVUE2 Figure 5.21: Weibull Coverage Probabilities vs. Width for Case C: a 2 varies 62

78 5.2 Hypothesis Testing Results and Discussion For conciseness, only selected results are presented here. The rest of the results are available in Appendix F. We use the same set of parameters from Tables for the simulation. The Type I and power rates are used to assess the performance of the different test procedures. Type I (when Ho is true) and power rates (when Ho is not true) are computed as the relative frequency of the number of times Ho is rejected, i.e. # (reject H 0 ) / We want the Type I error rates to be close to 0.05 and the power rate to be as high as possible On A-distribution: Results and Discussion The results for cases where the proportions of zeros are unequal are presented in Table 5.13 (selected cases only). In general, the two-part tests and MLEs control the Type I error best. The rest of the methods have good Type I error control. MVUE2 and R-MVUE2 are a little conservative, i.e. they reject H 0 less than the expected Since all the methods have good Type I error control we can now assess the power of each method. We notice that the two-part methods by Lachenbruch perform best. As the differences in the proportions of zeros increases between the two groups, or consequently as the absolute difference (ki k, 2 ) increases, the powers of the two-part tests, T-P t and T-P w, increase remarkably. This is due to the fact that zeros are explicitly handled in the two-part models. As anticipated, the power of all the methods gets better as the sample sizes increase. For a graphical comparison of all methods, the power curves of the tests at sample sizes of 50 are shown in Figure For cases where the means vary, results are shown in Table In terms of controlling the Type I error, the performance of all the methods are similar to

79 cases where only the proportions vary. The two-part tests and MLEs still have the best Type I error control. MVUEs and all the robust tests have good Type I error control but are conservative. In terms of power of the tests, T-P w performs best while the proposed robust tests, R-BC and R-TB, perform very well for cases where the difference (ki k 2 ) are moderate to large. The latter tests provide some improvements from their regular MLE counterparts. The power curves of these tests are shown in Figure 5.23 for a better assessment of their performance. In assessing the performance of the tests on cases where the a 2 varies, we look at the simulation results in Table We notice that the ts and MLEs perform best in controlling the Type I error for all values of n. The other methods have good error rate control as well. All the robust methods though are a little conservative in controlling the type I error especially for 5 of 0.1, but the Type I error control gets better for larger values of 5 (0.2 and 0.4). Looking at the power of the different tests that have acceptable error rates, the MLEs perform best followed by MVUE1 and the ts. The results can also be seen in the power curves of the tests shown in Figure

80 Table 5.13: Power Rates of Tests for Equal /i and a 2 Cases (Kl - fc2) Method n a6 a7 a8 a9 alo t tw W MVUE MVUE R-MVUE R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUE MVUE R-MVUE R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUE MVUE R-MVUE R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB

81 10% zero values 20% zero values 40% zero values Figure 5.22: Power curves for Case A: 5 varies (equal fi and a 2 ), at n 66

82 Table 5.13: Power Rates of Tests for Equal /i and a 2 Cases (KI - k 2) Method n b6 b7 b8 b9 blo t tw W MVUEl MVUE R-MVUE R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUE R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUE R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB

83 10% zero values 20% zero values 40% zero values n r "i 1 1 r mean difference -e- t -A- tw -*- T-Pt mean difference mean difference + W MVUE1 MVUE2 R-MVUE1 R-MVUE2 T-Pw MLE-BC-O- MLE-TB» R-BC -H- R-TB Figure 5.23: Power curves for Case B: ji varies (equal 5 and a 2 ), at n = 50 68

84 Table 5.13: Power Rates of Tests for Equal /i and a 2 Cases («i - k 2) Method n c6 c7 c8 c9 clo t tw W MVUEl MVUE R-MVUE R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUE R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUE R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB

85 10% zero values 20% zero values 40% zero values n 1 r "i i i r mean difference -A- tw -*- T-Pt mean difference mean difference + W X- MVUE1 MVUE2 R-MVUE1-B- R-MVUE2 0 T-Pw «MLE-BC-S- MLE-TB-B- R-BC -H- R-TB Figure 5.24: Power curves for Case C: a 2 varies (equal 5 and p), at n 70

86 5.2.2 On Gamma and Weibull Distributions: Results and Discussion Results on Type I error and power rates are shown in Appendix G for gamma distribution and Appendix H for Weibull distribution. On Gamma distribution for Case A where only the 5 varies, the two-part tests, wilcoxon and t control the type I error best. The other methods also have acceptable error control except for the MVUEs which are very conservative. Among the methods with good error control, the two-part tests are most powerful overall as seen in Figure Wilcoxon also has satisfactory power. For cases where only the /j varies (Case B), same set of methods as Case A control the type I error well. But the ts are most powerful overall. Methods with good power are T-P t, R-BC and R-TB. Lastly, for Case C where a 2 varies, all methods control the type I error well but the robust methods are conservative. Among the methods, the MLE's are most powerful. For large difference of a 2 in between the two groups, T-P w and W have good power as well. On Weibull distribution for Case A, the two-part tests and t control the type I error rate best, while other methods have good error control. The MVUEs are conservative though. The two-part tests are most powerful among all the methods, while W is next for cases where the difference of a 2 is large. For Case B, t control the type I error best. The rest of the methods control the error well but MVUE and R-MVUE2 are conservative. In terms of power, results are similar to Gamma distribution. That is, the ts are most powerful overall and T-P t, R-BC and R-TB have good power as the difference in /j between the two groups increases. Lastly, Case C also has similar results to gamma where all methods control the type I error well but the robust methods are conservative. Among the methods, the MLEs are

87 most powerful and ts have satisfactory power for large difference in a 2 between the two groups. Case A Case B CaseC i i i i r mean difference -e- t -i 1 1 i r mean difference - tw + W MVUE1-0- MVUE2 -Pt T-Pw MLE-BC-S- MLE-TB-B- R-BC "l I I I T mean difference R-MVUE1-B- R-MVUE2 -»- R-TB Figure 5.25: Power curves for Gamma Distribution, at n = 50 72

88 Case A Case B Case C t r T mean difference mean difference mean difference -a- t tw + W x- MVUE1 MVUE2 R-MVUE1-B- R-MVUE2 -*- T-Pt T-Pw MLE-BC-S- MLE-TBR-BC Figure 5.26: Power curves for Weibull Distribution, at n = 50 73

89 5.3 Wilcoxon-based Simulation and Results This section aims to elaborate more on the performance of the Wilcoxon-based interval on three different parameters on measuring the difference between two A-distributions: (1) difference of means, (2) difference of medians, and (3) median of differences. Using the parameter combination in Table 4.1, selected results are presented here. Type I error rates or power rates of the test (depending on whether H 0 is true or not) are the same for all three parameters, same with the average width of the interval. Thus, we will only compare the coverage probabilities, coverage error, upper error rate and lower error rate of the three parameters for the difference. Tables 5.16 and 5.17 show results for cases where the S varies, while Tables 5.18 and 5.19 present the cases where the fi differs for both groups. And lastly, Tables 5.20 and 5.21 are for cases where the a 2 varies. The following are the highlights of the simulation: Wilcoxon-based interval measures the median of the differences best. The coverage probability for all the results are closest to nominal level of Also, this parameter has the most balanced error rates. Consequently, true values of the difference of means (ki k 2 ) or difference of medians (mi 7712) that are closest to the median of differences m have better coverage probabilities. For cases where the true median of differences m is zero (the other two parameters have true difference other than zero), we notice that under "power", the values are very low and essentially close to 0.05, more appropriate for a Type I error rate.

90 Table 5.16: Case 3: Results for varying <5 (equal /z and a 2 ) Si Kl - K2 m l ~?7l2 TO nl=n2 Parameter Power CP CE LER UER AW Table 5.17: Case 5: Results for varying S (equal /i and a 2 ) Si $2 M a 2 Kl - K2 m l ~ m 2 TO nl=n2 Parameter Power CP CE LER UER AW

91 Table 5.18: Case 8: Results for varying /v. (equal 5 and a 2 ) Ml M2 5 K\ - K2 mi 7712 m nl=n2 Parameter Power CP CE LER UER AW Table 5.19: Case 10: Results for varying //, (equal 5 and a 2 ) Ml M2 8 a 2 /Cl K2 mi m 2 m nl=n2 Parameter Power CP CE LER UER AW

92 Table 5.20: Case 13: Results for varying a 2 (equal <5 and /j ) 4 8 Kl - mi TO nl=n2 Parameter Power CP CE LER UER AW Table 5.21: Case 15: Results for varying a 2 (equal 8 and // ) 8 M Kl K2 mi m nl=n2 Parameter Power CP CE LER UER AW

93 To reassess earlier simulation results on Wilcoxon under A-distribution, we look at the performance of Wilcoxon given that it measures the median of differences (W(new)). To compare W(new) to other methods, an example is shown in Tables The improvement in coverage was very noticeable. In fact in all cases, given that Wilcoxon measures the right parameter (median of differences), it is the best method with the best coverage and narrowest width in comparison with the rest of the methods used in the simulation study. Table 5.22: 95% CI under Ai(0.1, 0.5, 1) and A 2 (0.5, 0.5, 1) ki = , k 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W(new) MVUE MVUE R-MVUE R-MVUE t tw W(new) MVUE MVUE R-MVUE R-MVUE t tw W(new) MVUE MVUE R-MVUE R-MVUE

94 Table 5.23: 95% CI under A^O.2, 0, 1) and A 2 (0.2, 0.9, 1) ki = , K 2 = , - K 2 = Method Sample Size CP CE LER UER Width t tw W(new) MVUEl MVUE R-MVUE R-MVUE t tw W(new) MVUEl MVUE R-MVUE R-MVUE t tw W(new) MVUEl MVUE R-MVUE R-MVUE

95 Table 5.24: 95% CI under Ai(0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 2) K x = , k 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W(new) MVUE MVUE R-MVUE R-MVUE t tw W(new) MVUE MVUE R-MVUE R-MVUE t tw W(new) MVUE MVUE R-MVUE R-MVUE

96 Chapter 6 CONCLUSIONS AND FUTURE RESEARCH There are limited interval estimators for the difference of means of two delta distributions available in literature. Classical methods like Student's t and Wilcoxonbased methods are still widely used even for skewed data and data inflated with zero values. Moreover, existing MVUE intervals on delta distribution have satisfactory performance only when there is low skewness in the data. These MVUE methods are functions of classic estimators x and a 1 which are not robust. To make these MVUE methods less susceptible to the effects of high variability in the data, a direct substitution of x and cr 2 with more robust estimators, like Th and S&, can be done. This straightforward procedure has improved the methods in terms of coverage properties and interval widths. In the simulation study, R-MVUE1 specifically demonstrated excellent performance for most of the cases under delta distribution. Even when the positive part of the data came from a gamma or Weibull distribution, R-MVUE1 showed resilience by still having good coverage and narrow interval width. But surprisingly, t and tw have better performance under this deviation from the delta model. The ts only break down when variability on both groups are high, which is the same for the other methods considered.

97 For testing the difference between two delta means, different methods were considered. The testing were done through confidence intervals except for the twopart tests by Lachenbruch (2001) which were based on test statistics. A simulation was done to assess the Type I error and power rates of the tests. The proposed robust methods R-BC and R-TB showed some improvements from their regular counterparts MLE-BC and MLE-TB, respectively, in most of the cases. For any distribution (delta, gamma or Weibull), the two-part tests are most powerful in situations where the difference in means (k\ k 2 ) is due to the difference in <5 between the two groups. When the difference in means is due to the varying values of /i. T-P w is most powerful under delta distribution. But R-BC and R-TB have very satisfactory power as well. For gamma and Weibull though, the ts are most powerful. Lastly, if the difference in means is due to the unequal a 2 between the two groups, the MLE-BC and MLE-TB are the most powerful methods. The difference in means is not measured well by the Wilcoxon-based interval. It gave the narrowest intervals on almost all the cases in the simulation study but the coverage probabilities are very low. To measure the difference of the two delta distributions using the Wilcoxon-based interval, we compared three parameters of difference namely, difference of means, difference of medians and median of differences. Among the three, it was found that the Wilcoxon-based interval mesure the median of differences best. For future research, proposed methods could be used to test difference in means of two zero-inflated distributions through test statistics that improve both the Type I error and power rates of the tests. This could also be extended to censored data. Other techniques like bootstrap on both intervals and tests could also be explored. 82

98 Appendix A Confidence Intervals for A-distribution Case A 83

99 Table A.l: 95% CI under A^O.l, 0.5, 1) and A 2 (0.1, 0.5, 1) /ci = , K 2 = , ki - k 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE

100 Table A.2: 95% CI under A^O.l, 0.5, 1) and A 2 (0.2, 0.5, 1) Kl = , k 2 = , - K2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUE R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

101 Table A.3: 95% CI under A^O.l, 0.5, 1) and A 2 (0.3, 0.5, 1) Kl = , k 2 = , k x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE

102 Table A.4: 95% CI under A^O.l, 0.5, 1) and A 2 (0.4, 0.5, 1) Kl = , k 2 = , k x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

103 Table A.5: 95% CI under A^O.l, 0.5, 1) and A 2 (0.5, 0.5, 1) Ki = , k 2 = , m - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

104 Table A.6: 95% CI under Ai(0.3, 0.5, 1) and A a (0.3, 0.5, 1) Ki = , k 2 = , - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

105 Table A.7: 95% CI under Ai(0.3, 0.5, 1) and A 2 (0.4, 0.5, 1) «i = , k 2 = , ki - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE

106 Table A.8: 95% CI under Ax (0.3, 0.5, 1) and A 2 (0.5, 0.5, 1) K\ = , K 2 = , - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

107 Table A.9: 95% CI under Ai(0.3, 0.5, 1) and A 2 (0.6, 0.5, 1) ki = , k 2 = , m - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE

108 Table A.10: 95% CI under Ai(0.3, 0.5, 1) and A 2 (0.7, 0.5, 1) ki = , k , - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

109 Appendix B Confidence Intervals for A-distribution Case B 94

110 Table B.l: 95% CI under Ai(0.1, 0, 1) and A 2 (0.1, 0, 1) ki = , k>2 = , «i - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE

111 Table B.2: 95% CI under A^O.l, 0, 1) and A 2 (0.1, 0.3, 1) K X = , K 2 = , ki - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

112 Table B.3: 95% CI under Ai(0.1, 0, 1) and A 2 (0.1, 0.5, 1) ki = , k 2 = , /c x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE

113 Table B.4: 95% CI under A^O.l, 0, 1) and A 2 (0.1, 0.7, 1) K\ = , K 2 = , ki - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

114 Table B.5: 95% CI under A^O.l, 0, 1) and A a (0.1, 0.9, 1) ki = , K 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

115 Table B.6: 95% CI under Ax (0.4, 0, 1) and A 2 (0.4, 0, 1) ki = , K 2 = , Kl - k 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

116 Table B.7: 95% CI under Ai(0.4, 0, 1) and A 2 (0.4, 0.3, 1) ki = , k 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE t tw W MVUEl MVUE R-MVUE R-MVUE

117 Table B.8: 95% CI under A x (0.4, 0, 1) and A 2 (0.4, 0.5, 1) ki = , K 2 = , - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

118 Table B.9: 95% CI under Ai(0.4, 0, 1) and A 2 (0.4, 0.7, 1) ki = , k 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

119 Table B.10: 95% CI under Ai(0.4, 0, 1) and A 2 (0.4, 0.9, 1) ki = , k 2 = , ki - k 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

120 Appendix C Confidence Intervals for A-distribution Case C 105

121 Table C.l: 95% CI under Ai(0.1, 0.5, 0.15) and A 2 (0.1, 0.5, 0.15) ki = , k 2 = , «i - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

122 Table C.2: 95% CI under A^O.l, 0.5, 0.15) and A 2 (0.1, 0.5, 0.5) ^ = , K 2 = , - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

123 Table C.3: 95% CI under A^O.l, 0.5, 0.15) and A 2 (0.1, 0.5, 1) K\ = , «2 = , Kl - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

124 Table C.4: 95% CI under A 2 (0.1, 0.5, 0.15) and A 2 (0.1, 0.5, 1.5) «i = , = , m - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUE MVUE R-MVUEl R-MVUE

125 Table C.5: 95% CI under A 1 (0.1, 0.5, 0.15) and A 2 (0.1, 0.5, 2) ki = , K 2 = , Kl - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

126 Table C.6: 95% CI under A^CU, 0.5, 0.15) and A 2 (0.4, 0.5, 0.15) K\ = , K2 = , - k 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUE MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

127 Table C.7: 95% CI under Ai(0.4, 0.5, 0.15) and A 2 (0.4, 0.5, 0.5) «i = , k 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

128 Table C.8: 95% CI under Ai(0.4, 0.5, 0.15) and A 2 (0.4, 0.5, 1) ki = , «2 = , ki - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

129 Table C.9: 95% CI under Ai(0.4, 0.5, 0.15) and A 2 (0.4, 0.5, 1.5) «i = , k 2 = , k x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

130 Table C.10: 95% CI under A x (0.4, 0.5, 0.15) and A 2 (0.4, 0.5, 2) ki = , k 2 = , k x - k 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

131 Appendix D Confidence Intervals for A-distribution with 100% Contamination from Gamma Distribution 116

132 Table D.l: 95% CI under Ai(0.2, 0.5, 1) and A 2 (0.2, 0.5, 1) with 100% Contamination from Gamma Distribution ki = , K 2 = , m - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+05 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl ~ R-MVUE

133 Table D.2: 95% CI under Ai(0.2, 0.5, 1) and A 2 (0.3, 0.5, 1) with 100% Contamination from Gamma Distribution ki = , k 2 = , K x - k 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+13 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

134 Table D.3: 95% CI under A x (0.2, 0.5, 1) and A 2 (0.4, 0.5, 1) with 100% Contamination from Gamma Distribution ki = , k 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+07 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

135 Table D.4: 95% CI under Ai(0.2, 0.5, 1) and A 2 (0.5, 0.5, 1) with 100% Contamination from Gamma Distribution ki = , K 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+10 R-MVUEl R-MVUE t tw W MVUEl MVUE E+07 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

136 Table D.5: 95% CI under Ax (0.2, 0.5, 1) and A 2 (0.6, 0.5, 1) with 100% Contamination from Gamma Distribution ki = , K 2 = , Kl - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+11 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

137 Table D.6: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0, 1) with 100% Contamination from Gamma Distribution AC! = , K 2 = , ki - k 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+10 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

138 Table D.7: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.3, 1) with 100% Contamination from Gamma Distribution ki = , K 2 = , ki - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

139 Table D.8: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.5, 1) with 100% Contamination from Gamma Distribution ki = , k 2 = , K x - k 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+05 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

140 Table D.9: 95% CI under Ax(0.2, 0, 1) and A 2 (0.2, 0.7, 1) with 100% Contamination from Gamma Distribution «! = , k 2 = , m - k 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+08 R-MVUEl R-MVUE t tw W MVUEl MVUE E+03 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

141 Table D.10: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.9, 1) with 100% Contamination from Gamma Distribution m = , «2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+05 R-MVUEl R-MVUE t tw W MVUEl MVUE E+03 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

142 Table D.ll: 95% CI under A x (0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 0.15) with 100% Contamination from Gamma Distribution ki = , k 2 = , KI - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

143 Table D.12: 95% CI under A x (0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 0.5) with 100% Contamination from Gamma Distribution ki = , ac 2 = , - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

144 Table D.13: 95% CI under A^O.2, 0.5, 0.15) and A 2 (0.2, 0.5, 1) with 100% Contamination from Gamma Distribution m = , K 2 = , m - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE E+04 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

145 » Table D.14: 95% CI under A x (0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 1.5) with 100% Contamination from Gamma Distribution ki = , k 2 = , m ~ K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl E+04 MVUE E+26 R-MVUEl R-MVUE t tw W MVUEl E+04 MVUE E+13 R-MVUEl R-MVUE t tw W MVUEl E+05 MVUE E+11 R-MVUEl R-MVUE

146 Table D.15: 95% CI under Ai(0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 2) with 100% Contamination from Gamma Distribution Ki = , k 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl E+16 MVUE E+111 R-MVUEl R-MVUE t tw W MVUEl E+13 MVUE E+48 R-MVUEl R-MVUE t tw 0: W MVUEl E+19 MVUE E+50 R-MVUEl R-MVUE

147 Appendix E Confidence Intervals for A-distribution with 100% Contamination from Weibull Distribution 132

148 Table E.l: 95% CI under Ai(0.2, 0.5, 1) and A 2 (0.2, 0.5, 1) with 100% Contamination from Weibull Distribution ki = , k 2 = , - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

149 Table E.2: 95% CI under Ai(0.2, 0.5, 1) and A 2 (0.3, 0.5, 1) with 100% Contamination from Weibull Distribution ki = , k 2 = , m - k 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

150 Table E.3: 95% CI under Ax(0.2, 0.5, 1) and A 2 (0.4, 0.5, 1) with 100% Contamination from Weibull Distribution = , k 2 = , Kl- K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

151 Table E.4: 95% CI under Ai(0.2, 0.5, 1) and A 2 (0.5, 0.5, 1) with 100% Contamination from Weibull Distribution /ci = , k 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

152 Table E.5: 95% CI under Ai(0.2, 0.5, 1) and A 2 (0.6, 0.5, 1) with 100% Contamination from Weibull Distribution ki = , K 2 = , Kl - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+06 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

153 Table E.6: 95% CI under Ax(0.2, 0, 1) and A 2 (0.2, 0, 1) with 100% Contamination from Weibull Distribution ki = , k 2 = , - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

154 Table E.7: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.3, 1) with 100% Contamination from Weibull Distribution ki = , K 2 = , KI - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+02 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

155 Table E.8: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.5, 1) with 100% Contamination from Weibull Distribution Ki = , k 2 = , m - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

156 Table E.9: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.7, 1) with 100% Contamination from Weibull Distribution m = , K 2 = , m - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+05 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

157 Table E.10: 95% CI under Ai(0.2, 0, 1) and A 2 (0.2, 0.9, 1) with 100% Contamination from Weibull Distribution ki = , K 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

158 Table E.ll: 95% CI under Ai(0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 0.15) with 100% Contamination from Weibull Distribution KI = , K 2 = , ki - K 2 = 0 Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

159 Table E.12: 95% CI under A x (0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 0.5) with 100% Contamination from Weibull Distribution «i = , k 2 = , K x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

160 Table E.13: 95% CI under A^O.2, 0.5, 0.15) and A 2 (0.2, 0.5, 1) with 100% Contamination from Weibull Distribution ki = , K 2 = , k x - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

161 Table E.14: 95% CI under Ai(0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 1.5) with 100% Contamination from Weibull Distribution ki = , k 2 = , Ki - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+08 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

162 Table E.15: 95% CI under Ai(0.2, 0.5, 0.15) and A 2 (0.2, 0.5, 2) with 100% Contamination from Weibull Distribution ki = , k 2 = , KI - K 2 = Method Sample Size CP CE LER UER Width t tw W MVUEl MVUE E+09 R-MVUEl R-MVUE t tw W MVUEl MVUE E+07 R-MVUEl R-MVUE t tw W MVUEl MVUE R-MVUEl R-MVUE

163 Appendix F Type I Error and Power Rates for A-distribution 148

164 Table 5.13: Power Rates of Tests for Equal /i and a 2 Designs (ki - K2) Method n al a2 a3 a4 a5 t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB

165 Table F.2: Power Rates of Tests for Equal //. and a 2 Designs (ki - K 2 ) Method n all al2 al3 al4 al5 t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB

166 Table 5.13: Power Rates of Tests for Equal /i and a 2 Designs («l - k 2) Method n bl b2 b3 b4 b5 t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB

167 Table 5.13: Power Rates of Tests for Equal /i and a 2 Designs («1 - K 2 ) Method n bll bl2 bl3 bl4 bl5 t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB t tw W MVUEl MVUE R-MVUEl R-MVUE T-P t T-P w MLE-BC MLE-TB R-BC R-TB