Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Size: px
Start display at page:

Download "Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at"

Transcription

1 Some Applications of Exponential Ordered Scores Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 26, No. 1 (1964), pp Published by: Wiley for the Royal Statistical Society Stable URL: Accessed: :49 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Royal Statistical Society, Wiley are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. Series B (Methodological)

2 1964] 103 Some Applications of Exponential Ordered Scores By D. R. Cox Birkbeck College, University of London [Received August Revised September 1963] SUMMARY There are simple standard tests for the comparison of samples assumed to arise from exponential distributions, but the properties of these tests are known to depend severely on the assumption of exponential form. However, the observations can sometimes usefully be ranked, and then replaced by the corresponding expected values of order statistics in sampling the unit exponential distribution, before calculating the appropriate test statistic. Some special tests of this type are examined. The procedure is analogous to the use of Fisher and Yates's scores in normal theory. Savage (1956) gave the test of this type for comparing two samples, in the course of a general study of rank tests for the two-sample problem. 1. INTRODUCTION IN 1938, in the first edition of their Statistical Tables, Fisher and Yates gave the expected values of the order statistics in sampling a standardized normal distribution, the so-called normal scores (Fisher and Yates, 1957, Table 20). They suggested that these scores should be used for significance tests involving comparisons with ranked data, assumed to be derived from an underlying normal distribution. For example, to compare the locations of two samples, the ranks are replaced by the corresponding normal scores and, as an approximation, the standard t-test is then applied. Similar methods apply for testing null hypotheses about problems of regression, serial correlation, etc. involving ranks. The intuitive basis of these procedures is, of course, that the normal scores provide a good reconstruction of the underlying variate values on a standardized scale, the null hypothesis being assumed true. Much later, mdre formal justifications of the use of normal scores were given. The point of view taken in this work is that, although the variate values may be available, it is required to test a non-parametric null hypothesis with a rank-invariant test. In particular, Terry (1952) showed that among all rank-invariant tests for the two-sample location problem the one based on normal scores is locally most powerful for detecting a difference in the means of two normal populations. Chernoff and Savage (1958) showed that for detecting shifts in location of an arbitrary distribution, the test based on normal scores has asymptotically a power greater than or equal to that of the usual large-sample normal test using variate values. In particular, if the populations are normal there is asymptotically no loss of power from using normal scores rather than variate values. Lehmann (1959, pp ) has discussed this work. The use of normal scores is only a special case of a general treatment. The above application of expected order statistics is distinct from that to the analysis of censored data; see, especially, Plackett (1958, 1959) for such use both for normal and logistic distributions.

3 104 Cox - Some Applications of Exponential Ordered Scores [No. 1, Now there has been much recent interest in statistical methods associat the exponential distribution, for instance in connection with application testing; see, for example, Maguire et al. (1952), Epstein and Sobel (1953) and Birn (1954). The simplicity of these methods, and the ease with which they are exten censored data, make them attractive, but unfortunately the methods are s to the assumption of exponential form (Zelen and Dannemiller, 1961). Th taken with the properties sketched above of procedures based on normal suggests the following procedure for significance tests involving the comp exponential distributions: rank the observations, replace the ranks by the c sponding expected order statistics in sampling the unit exponential distribu then calculate the usual exponential theory test statistic. In this way we ca to lose very little when, in fact, the underlying distribution is close to the exp form, while retaining control over the test if there is an appreciable departure the exponential distribution. The special case of this procedure for the comparison of two samples was given by Savage (1956) in the course of a general study of rank tests for the two-sample problem. The formal justification for the procedure, contained in the general work referred to above, is strong. When the true distributions are exponential, the test is a locally most powerful rank test with no asymptotic loss of power. If the true distributions are not exponential, the exact exponential theory test will, except in very special cases, be invalid, whereas the test based on exponential scores is distribution-free. It seems very likely that the same justification applies not only to the two-sample problem, but also to the other applications of this paper. Unfortunately, the loss of information in small samples when the true distribution is exponential, while very probably small, is not known. The present paper investigates some tests based on exponential scores. Of course, in practice, formal tests of significance are likely to be only part of the whole analysis. An important thing, for instance, is likely to be graphical analysis of the survivor curves of the separate sets of data. 2. THE EXPONENTIAL SCORES In this Section we examine some properties of the finite population consisting of the expected order statistics for a random sample of size n from the distribution e-v. Denote by trn the expected value of the rth observation in increasing order of magnitude. It is well known that so that in particular trn = /n+ + 1/(n-r+ 1) (r =1,...,n), (1) tin = 1/n, t2n = 1/n + 1/(n- 1), (2) tnn = 1/n +11(n -1)+%+ 1. For n not too large, these are easily The moments of the finite populatio sums n Sin = trinn r=1

4 1964] Cox - Some Applications of Exponential Ordered Scores 105 Now the triangular scheme (2) for sample size n is derived from that for sample size n -1 by adding the first column of 1/n's. There follow easily recurrence relations such as the following: 51n = 51,n-1 + 1, S2n = 52,n-1 + (2/n) s1,n-1 + 1/n, S3n = S3,n-1 + (3/n) S2,n-1 + (3/n2) s1,n-1 + I /ni J These can be solved subject to the initial condition sil = 1. In particular, n Sin = n., S2n = 2n - (I /r) = 2n -tnn, r=1 After some calculation, we have, for Fisher's K parameters for the finite population, Kln= 1, K27, = nn-tn = 1 logn+y-1 0(lo( Kln 2n -n-i n n where y = is Euler's constant. The leading term in the asymptotic expansion for each cumulant is the corresponding cumulant of the unit exponential distribution. Table 1 gives some values of K2n and of the approximation 1- (log n + y - 1)/n. TABLE 1 Some constants connected with the exponential scores n K2n K2 n ln ln (approx.) (approx.) P f P f P P132 1P f863 0f f897 0f901 1P082 1P P067 1P f930 1P057 1f COMPARISON OF Two SAMPLES Suppose that we have two independent samples of sizes ml, m2 with sample means il, Y2. To examine the null hypothesis that the population means are equal, assumin the distributions to be exponential, we test j13/2 in the F-distribution with (2ml, 2m2) degrees of freedom. Now follows the procedure sketched in Section 1. Let the whole set of n = ml+ m2 values be ranked and replaced by the scores {trn} o We assume for simplicity that there are no ties. Let 4l., T2 be the sample mean scores. A first approximation is then to test [1/12, again as F with (2ml, 2m2) degrees of freedom. The "exact" test procedure, however, is to regard the set {trn} of scores as a finite population. Under the null hypothesis, the scores arising in, say, the first sample are a random sample of ml drawn without replacement from the finite population. Since ml 1 + m2 t2 = Sin and is fixed, a critical region based, say, on large i1/i2 is equivalent to one based on large ii., or, of course, on ul = ml f1/s1n. Hence we can regard the test as based on one sample mean.

5 106 Cox - Some Applications of Exponential Ordered Scores [No. 1, From the theory of sampling a finite population (Kendall and Stuart, 1958, pp. 301, 302), we have, from (3) and (4), that under the null hypothesis E(ul) = ml/n, var (ul) = (ml M2) (n - t..)/{n3(n - 1)}. (5 Now the simplest test based on ul would be to apply the F test to i1/t2, just as if we were working with the original observations. This is equivalent to treating ul having the beta distribution xmi-l(1 - x)m2-'/b(ml, i2). (6) For this the first two moments corresponding to (5) ar ml/(mi+m2) and mlm2/{(ml?m?)2 A natural modification to improve the agreement T-/T2 as having an F-distribution with (21ml,21m2 Corresponding to (7), we have a mean and variance ml/(ml + M2) and ml m2/{(ml + m2)2 (Ml + 1m and these agree with (5) if I is chosen as a function 1~n-2n+t ~~ y-r2+logn (log n In =n( _n - tnn) + n +O( kln). (8) The last two columns of Table 1 give some values for the factor In approximation 1 + (y -2 + log n)/n. We could test the above procedure by finding higher moments of the te Instead a simple example has been worked out in detail. Example. Suppose that we have for comparison two samples of five o each. The first step is to rank the whole set of ten observations and t ranks by the scores (2), namely 0.10, 021, 034, 0-48, 065, 0O85, 1.10, 1P43, 1P93, We now calculate the sample mean scores tl, i2 and base our test on -l/i2 lently, on i1. We consider four methods of computing significance limits. Method 1. There is an "exact" permutation test, since under the null the first sample is equally likely to be any of the 252 distinct sets of 5 num can be formed from the finite population of 10. The resulting distribution been enumerated, with a grouping interval of 0-1, and interpolated va 25, 10, 5, 2i upper per cent. points are given in Table 2. The dist symmetrical with mean 5, so that the lower tail need not be examined. Method 2. We can use a crude approximation treating the exponen as variate values. Then F-= T1/T2 = T1/(2 - i1) is tested in the variancebution with (10, 10) degrees of freedom. Resulting values of 5T1 = 1O given in Table 2. Method 3. This is the same as Method 2 except that modified degrees of freedom are used in accordance with (8). Since 11 = 1-173, the new degrees of freedom are (11.7, 11.7). Table 2 gives the revised limits for 5T1.

6 1964] Cox - Some Applications of Exponential Ordered Scores 107 Method 4. We can treat 5i1 as normally distributed with the mean and variance of the exact permutation distribution. The method is a very special case of (11) of the next Section, in which v = 5!1 and xl =... = X5 = 1, X6 =... = X10 =O. approximation is to treat 54- as normally distributed with mean 5 and standard deviation 1(10K10/4) = 1P40. TABLE 2 Comparison of significance limits for a two-sample problem Probability "Exact" Crude F Modified F Normal level limit approximation approximation approximation per cent. (Method 1) (Method 2) (Method 3) (Method 4) j The general conclusion from Table 2 is that even with these extremely small sample sizes, both F approximations and the normal approximation are reasonably good and that in the tail there is a worth-while improvement from using the F-distribution with the correcting factor I.. It is not reasonable in this example to expect the approximations to apply much beyond the range of Table 2, since there are only 252 possible distinct samples. However, the extreme tail can easily be found exactly. Thus the two highest values of 5 il are * = 8-24, P = 8&04, each with probability 1/252. Thus the value attained or exceeded with probability 1/126 is (The approximate 1 per cent. point from the crude F approximation is 8-29, and 8 10 for the modified approximation.) TABLE 3 Comparison of significance limits for total offirst sample, for two-sample problem with sample sizes 3, 9 Probability "Exact" Crude F Modified F Normal level limit approximation approximation approximation per cent. (Method 1) (Method 2) (Method 3) (Method 4) Lower * P Upper *88 6*

7 108 Cox - Some Applications of Exponential Ordered Scores [No. 1, A more severe test of the approximations is provided by a problem with very unequal sample sizes. Table 3 summarizes some calculations on the comparison of two samples of sizes 3, 9, for which the permutation distribution has 210 points. On the whole, there is a substantial improvement from using the correcting factor I., especially in the tails. The normal approximation makes no allowance for the skewness of the distribution of the first sample total and is rather poor. It is very reasonable to expect that as sample sizes increase and all the distributions get more nearly normal, the two approximations provided by the F-distribution will improve. The two specimen examples involve very small sample sizes, and the satisfactory approximations obtained in Tables 2 and 3 suggest that the approximations hold for all reasonable practical purposes. 4. REGRESSION ANALYSIS Suppose that we want to test the observations y1,...,y for regression on the fixed quantities xi,..., x. It is worth outlining briefly the procedure to be followed if Yi'...,Yn are assumed exponentially distributed. In the absence of specific reason to the contrary, it is sensible to consider a simple regression model chosen for mathematical convenience. This is that the yj are independently exponentially distribute with means I/(oQ + fixj). The null hypothesis is that / = 0. The likelihood is exp (- oyyj - /xj yj) II (o + fixj), the sufficient statistic is (Zyj, Zxjyj) and the "exact" test is based on the conditi distribution of Ex1y1 given Xy, which is independent of the nuisance parameter oz. Vincent (1961) has discussed this test in the different context of tests on sample estimates of variance. An interesting special case arises when xj =j; if the xj are interpreted as intervals between events in a point process, the null hypothesis is that we have a Poisson process, whereas under the alternative process there is a trend with the serial number of the event. This can be compared with the model and test considered by Cox (1955) of a time-dependent Poisson process in which the probability rate of occurrence at time -r is oze8. The two significant tests of the null hypothesis /3 = 0 are almost identical; the models are, of course, different when go 0, although equivalent in a deterministic approximation. Another special case is the two-sample problem of Section 3 obtained when the xi takes values 1, 0 only. Turning now to the test based on scores, we rank the yj, replace them by the appropriate scores and consider the test statistic v = Xx. t., where the random variable tj is the score attached to the jth observation. Under the null hypothesis, all permutations of the scores {trn} are equally likely. We shall not develop the theory of v beyond the simplest normal approximation, for which we need the mean and variance of v over all permutations of {trn}. Now E(v) is bilinear in {xj} and {trn} and is symmetric in the two sets separately. Hence E(v) = a. sl 2xj =a. nyxj, where an depends only on n. Consideration of the special case xl =... = = 1 shows that a = 1/n and that E(v) = Zxj. (9)

8 1964] Cox - Some Applications of Exponential Ordered Scores 109 Similarly, we can show from considerations of degree and invariance that var (v) = bn 2(xj-x)2 K2, (10) where bn depends only on n. If we consider the we find on substituting into (10) that bn = 1. Thus the large-sample test is to take (v - Xj)I{(Xj-)2 -K2n}- (11) as normally distributed with mean zero and unit variance. In special cases, as in Section 3, it may be possible to obtain a better approximation fairly easily. 5. DisCUSSION There are a number of further points which will be discussed only very briefly. First, a number of further problems can be tackled by a direct extension of the above methods. One is the testing of the serial correlation coefficient. Secondly, if, say, in the two-sample problem both samples are censored at the same time point, we can adapt the above procedure in a simple way. If there are, say, p uncensored observations, we can attach scores tin,..., t1n to these as before, and the same score tp+l,n to all the censored observations. Now for exponentially distributed observations, the estimate of the population mean for the first population is (total time at risk)/(number of failures). Thus if there are r1 failures and ml-rl censored observations in the first sample, the corresponding statistic based on scores is {rl il + (ml - rl) t +1 n}/r1, (12) where il is the mean score of the failed individuals. The test sta two samples is the ratio of two statistics of the form (12). Approximately, this will have, under the null hypothesis, an F-distribution with (2r1, 2r2) degrees of freedom. Next, it is possible in principle to obtain confidence limits by the procedures of this paper. Thus in the two-sample problem of Section 3, we can test the hypothesis that the second population mean is po times the first population mean, for given po' by dividing the observations in the second sample by po before ranking and applying the test of Section 3. The set of all values po not rejected in such a significance test forms the required confidence interval. The practical procedure is to compute the standard normal deviate corresponding to the level of significance for a few trial values of po, and to interpolate to find the critical values determining the confidence limits. Finally, the procedures given here will have asymptotic optimum properties for the usual exponential theory alternatives, and therefore also for alternatives obtained by monotonic transformation from the exponential theory alternatives. In particular, the test of Section 3 is asymptotically optimum for testing the equality of two Weibull distributions, with distribution functions of the form 1-exp{-(o(x)f8}, when the alternative hypothesis is that the distributions have the same index f but different means.

9 110 Cox - Some Applications of Exponential Ordered Scores [No. 1, ACKNOWLEDGEMENT I am grateful to Miss B. M. Waters of the Scientific Computing Service, who did most of the calculations. REFERENCES BIRNBAUM, A. (1954), "Statistical methods for Poisson processes and exponential populations", J. Amer. statist. Ass., 49, CHERNOFF, H. and SAVAGE, I. R. (1958), "Asymptotic normality and efficiency of certain nonparametric test statistics", Ann. math. Statist., 29, Cox, D. R. (1955), "Some statistical methods connected with series of events", J. R. statist. Soc. B, 17, EPSTEIN, B. and SOBEL, M. (1953), "Life testing", J. Amer. statist. Ass., 48, FISHER, R. A. and YATES, F. (1957), Statistical Tables for Biological, Agricultural and Medical Research, 5th ed. Edinburgh: Oliver and Boyd. KENDALL, M. G. and STUART, A. (1958), Advanced Theory of Statistics, 1. London: Griffin. LEHMANN, E. L. (1959), Testing of Statistical Hypotheses. New York: Wiley. MAGUIRE, B. A., PEARSON, E. S. and WYNN, A. H. A. (1952), "The time intervals between industrial accidents", Biometrika, 39, PLACKETT, R. L. (1958), "Linear estimation from censored data", Ann. math. Statist., 29, (1959), "The analysis of life test data", Technometrics, 1, SAVAGE, I. R. (1956), "Contributions to the theory of rank order statistics: two-sample case, I", Ann. math. Statist., 27, TERRY, M. E. (1952), "Some rank order tests which are most powerful against specific parametric alternatives", Ann. math. Statist., 23, VINCENT, S. E. (1961), "A test of homogeneity for ordered variances", J. R. statist. Soc. B, 23, ZELEN, M. and DANNEMILLER, M. C. (1961), "The robustness of life testing procedures derived from the exponential distribution", Technometrics, 3,

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Biometrika Trust Some Simple Approximate Tests for Poisson Variates Author(s): D. R. Cox Source: Biometrika, Vol. 40, No. 3/4 (Dec., 1953), pp. 354-360 Published by: Oxford University Press on behalf of

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Biometrika Trust Robust Regression via Discriminant Analysis Author(s): A. C. Atkinson and D. R. Cox Source: Biometrika, Vol. 64, No. 1 (Apr., 1977), pp. 15-19 Published by: Oxford University Press on

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at American Society for Quality A Note on the Graphical Analysis of Multidimensional Contingency Tables Author(s): D. R. Cox and Elizabeth Lauh Source: Technometrics, Vol. 9, No. 3 (Aug., 1967), pp. 481-488

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at A Note on the Efficiency of Least-Squares Estimates Author(s): D. R. Cox and D. V. Hinkley Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 30, No. 2 (1968), pp. 284-289

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Regression Analysis when there is Prior Information about Supplementary Variables Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 22, No. 1 (1960),

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at On the Estimation of the Intensity Function of a Stationary Point Process Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 2 (1965), pp. 332-337

More information

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. Biometrika Trust An Improved Bonferroni Procedure for Multiple Tests of Significance Author(s): R. J. Simes Source: Biometrika, Vol. 73, No. 3 (Dec., 1986), pp. 751-754 Published by: Biometrika Trust Stable

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Biometrika Trust Some Remarks on Overdispersion Author(s): D. R. Cox Source: Biometrika, Vol. 70, No. 1 (Apr., 1983), pp. 269-274 Published by: Oxford University Press on behalf of Biometrika Trust Stable

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The Variance of the Product of Random Variables Author(s): Leo A. Goodman Source: Journal of the American Statistical Association, Vol. 57, No. 297 (Mar., 1962), pp. 54-60 Published by: American Statistical

More information

International Biometric Society is collaborating with JSTOR to digitize, preserve and extend access to Biometrics.

International Biometric Society is collaborating with JSTOR to digitize, preserve and extend access to Biometrics. 400: A Method for Combining Non-Independent, One-Sided Tests of Significance Author(s): Morton B. Brown Reviewed work(s): Source: Biometrics, Vol. 31, No. 4 (Dec., 1975), pp. 987-992 Published by: International

More information

NAG Library Chapter Introduction. G08 Nonparametric Statistics

NAG Library Chapter Introduction. G08 Nonparametric Statistics NAG Library Chapter Introduction G08 Nonparametric Statistics Contents 1 Scope of the Chapter.... 2 2 Background to the Problems... 2 2.1 Parametric and Nonparametric Hypothesis Testing... 2 2.2 Types

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. Selecting the Better Bernoulli Treatment Using a Matched Samples Design Author(s): Ajit C. Tamhane Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 42, No. 1 (1980), pp.

More information

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. Biometrika Trust A Stagewise Rejective Multiple Test Procedure Based on a Modified Bonferroni Test Author(s): G. Hommel Source: Biometrika, Vol. 75, No. 2 (Jun., 1988), pp. 383-386 Published by: Biometrika

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at The Interpretation of Interaction in Contingency Tables Author(s): E. H. Simpson Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 13, No. 2 (1951), pp. 238-241 Published

More information

A COMPARISON OF POISSON AND BINOMIAL EMPIRICAL LIKELIHOOD Mai Zhou and Hui Fang University of Kentucky

A COMPARISON OF POISSON AND BINOMIAL EMPIRICAL LIKELIHOOD Mai Zhou and Hui Fang University of Kentucky A COMPARISON OF POISSON AND BINOMIAL EMPIRICAL LIKELIHOOD Mai Zhou and Hui Fang University of Kentucky Empirical likelihood with right censored data were studied by Thomas and Grunkmier (1975), Li (1995),

More information

NAG Library Chapter Introduction. g08 Nonparametric Statistics

NAG Library Chapter Introduction. g08 Nonparametric Statistics g08 Nonparametric Statistics Introduction g08 NAG Library Chapter Introduction g08 Nonparametric Statistics Contents 1 Scope of the Chapter.... 2 2 Background to the Problems... 2 2.1 Parametric and Nonparametric

More information

HANDBOOK OF APPLICABLE MATHEMATICS

HANDBOOK OF APPLICABLE MATHEMATICS HANDBOOK OF APPLICABLE MATHEMATICS Chief Editor: Walter Ledermann Volume VI: Statistics PART A Edited by Emlyn Lloyd University of Lancaster A Wiley-Interscience Publication JOHN WILEY & SONS Chichester

More information

Research Article A Nonparametric Two-Sample Wald Test of Equality of Variances

Research Article A Nonparametric Two-Sample Wald Test of Equality of Variances Advances in Decision Sciences Volume 211, Article ID 74858, 8 pages doi:1.1155/211/74858 Research Article A Nonparametric Two-Sample Wald Test of Equality of Variances David Allingham 1 andj.c.w.rayner

More information

A nonparametric two-sample wald test of equality of variances

A nonparametric two-sample wald test of equality of variances University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 211 A nonparametric two-sample wald test of equality of variances David

More information

Testing Statistical Hypotheses

Testing Statistical Hypotheses E.L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Third Edition 4y Springer Preface vii I Small-Sample Theory 1 1 The General Decision Problem 3 1.1 Statistical Inference and Statistical Decisions

More information

The Periodogram and its Optical Analogy.

The Periodogram and its Optical Analogy. The Periodogram and Its Optical Analogy Author(s): Arthur Schuster Reviewed work(s): Source: Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character,

More information

NONPARAMETRICS. Statistical Methods Based on Ranks E. L. LEHMANN HOLDEN-DAY, INC. McGRAW-HILL INTERNATIONAL BOOK COMPANY

NONPARAMETRICS. Statistical Methods Based on Ranks E. L. LEHMANN HOLDEN-DAY, INC. McGRAW-HILL INTERNATIONAL BOOK COMPANY NONPARAMETRICS Statistical Methods Based on Ranks E. L. LEHMANN University of California, Berkeley With the special assistance of H. J. M. D'ABRERA University of California, Berkeley HOLDEN-DAY, INC. San

More information

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. On the Bound for a Pair of Consecutive Quartic Residues of a Prime Author(s): R. G. Bierstedt and W. H. Mills Source: Proceedings of the American Mathematical Society, Vol. 14, No. 4 (Aug., 1963), pp.

More information

FAILURE-TIME WITH DELAYED ONSET

FAILURE-TIME WITH DELAYED ONSET REVSTAT Statistical Journal Volume 13 Number 3 November 2015 227 231 FAILURE-TIME WITH DELAYED ONSET Authors: Man Yu Wong Department of Mathematics Hong Kong University of Science and Technology Hong Kong

More information

Ecological Society of America is collaborating with JSTOR to digitize, preserve and extend access to Ecology.

Ecological Society of America is collaborating with JSTOR to digitize, preserve and extend access to Ecology. Measures of the Amount of Ecologic Association Between Species Author(s): Lee R. Dice Reviewed work(s): Source: Ecology, Vol. 26, No. 3 (Jul., 1945), pp. 297-302 Published by: Ecological Society of America

More information

Applications of Basu's TheorelTI. Dennis D. Boos and Jacqueline M. Hughes-Oliver I Department of Statistics, North Car-;'lina State University

Applications of Basu's TheorelTI. Dennis D. Boos and Jacqueline M. Hughes-Oliver I Department of Statistics, North Car-;'lina State University i Applications of Basu's TheorelTI by '. Dennis D. Boos and Jacqueline M. Hughes-Oliver I Department of Statistics, North Car-;'lina State University January 1997 Institute of Statistics ii-limeo Series

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at A Renewal Problem with Bulk Ordering of Components Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 21, No. 1 (1959), pp. 180-189 Published by: Wiley

More information

Testing Statistical Hypotheses

Testing Statistical Hypotheses E.L. Lehmann Joseph P. Romano, 02LEu1 ttd ~Lt~S Testing Statistical Hypotheses Third Edition With 6 Illustrations ~Springer 2 The Probability Background 28 2.1 Probability and Measure 28 2.2 Integration.........

More information

Institute of Actuaries of India

Institute of Actuaries of India Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the

More information

Application of Parametric Homogeneity of Variances Tests under Violation of Classical Assumption

Application of Parametric Homogeneity of Variances Tests under Violation of Classical Assumption Application of Parametric Homogeneity of Variances Tests under Violation of Classical Assumption Alisa A. Gorbunova and Boris Yu. Lemeshko Novosibirsk State Technical University Department of Applied Mathematics,

More information

FULL LIKELIHOOD INFERENCES IN THE COX MODEL

FULL LIKELIHOOD INFERENCES IN THE COX MODEL October 20, 2007 FULL LIKELIHOOD INFERENCES IN THE COX MODEL BY JIAN-JIAN REN 1 AND MAI ZHOU 2 University of Central Florida and University of Kentucky Abstract We use the empirical likelihood approach

More information

ON SMALL SAMPLE PROPERTIES OF PERMUTATION TESTS: INDEPENDENCE BETWEEN TWO SAMPLES

ON SMALL SAMPLE PROPERTIES OF PERMUTATION TESTS: INDEPENDENCE BETWEEN TWO SAMPLES ON SMALL SAMPLE PROPERTIES OF PERMUTATION TESTS: INDEPENDENCE BETWEEN TWO SAMPLES Hisashi Tanizaki Graduate School of Economics, Kobe University, Kobe 657-8501, Japan e-mail: tanizaki@kobe-u.ac.jp Abstract:

More information

SEQUENTIAL TESTS FOR COMPOSITE HYPOTHESES

SEQUENTIAL TESTS FOR COMPOSITE HYPOTHESES [ 290 ] SEQUENTIAL TESTS FOR COMPOSITE HYPOTHESES BYD. R. COX Communicated by F. J. ANSCOMBE Beceived 14 August 1951 ABSTRACT. A method is given for obtaining sequential tests in the presence of nuisance

More information

SOME PROBLEMS CONNECTED WITH STATISTICAL INFERENCE BY D. R. Cox

SOME PROBLEMS CONNECTED WITH STATISTICAL INFERENCE BY D. R. Cox SOME PROBLEMS CONNECTED WITH STATISTICAL INFERENCE BY D. R. Cox Birkbeck College, University of London' 1. Introduction. This paper is based on an invited address given to a joint meeting of the Institute

More information

CHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007)

CHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007) FROM: PAGANO, R. R. (007) I. INTRODUCTION: DISTINCTION BETWEEN PARAMETRIC AND NON-PARAMETRIC TESTS Statistical inference tests are often classified as to whether they are parametric or nonparametric Parameter

More information

ON PITMAN EFFICIENCY OF

ON PITMAN EFFICIENCY OF 1. Summary ON PITMAN EFFICIENCY OF SOME TESTS OF SCALE FOR THE GAMMA DISTRIBUTION BARRY R. JAMES UNIVERSITY OF CALIFORNIA, BERKELEY A comparison is made of several two sample rank tests for scale change

More information

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika.

Biometrika Trust. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. Biometrika Trust Discrete Sequential Boundaries for Clinical Trials Author(s): K. K. Gordon Lan and David L. DeMets Reviewed work(s): Source: Biometrika, Vol. 70, No. 3 (Dec., 1983), pp. 659-663 Published

More information

Rank Regression with Normal Residuals using the Gibbs Sampler

Rank Regression with Normal Residuals using the Gibbs Sampler Rank Regression with Normal Residuals using the Gibbs Sampler Stephen P Smith email: hucklebird@aol.com, 2018 Abstract Yu (2000) described the use of the Gibbs sampler to estimate regression parameters

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at The Analysis of Exponentially Distributed Life-Times with Two Types of Failure Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 21, No. 2 (1959), pp.

More information

Distribution Fitting (Censored Data)

Distribution Fitting (Censored Data) Distribution Fitting (Censored Data) Summary... 1 Data Input... 2 Analysis Summary... 3 Analysis Options... 4 Goodness-of-Fit Tests... 6 Frequency Histogram... 8 Comparison of Alternative Distributions...

More information

n =10,220 observations. Smaller samples analyzed here to illustrate sample size effect.

n =10,220 observations. Smaller samples analyzed here to illustrate sample size effect. Chapter 7 Parametric Likelihood Fitting Concepts: Chapter 7 Parametric Likelihood Fitting Concepts: Objectives Show how to compute a likelihood for a parametric model using discrete data. Show how to compute

More information

Formulas and Tables by Mario F. Triola

Formulas and Tables by Mario F. Triola Copyright 010 Pearson Education, Inc. Ch. 3: Descriptive Statistics x f # x x f Mean 1x - x s - 1 n 1 x - 1 x s 1n - 1 s B variance s Ch. 4: Probability Mean (frequency table) Standard deviation P1A or

More information

Psychology 282 Lecture #4 Outline Inferences in SLR

Psychology 282 Lecture #4 Outline Inferences in SLR Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations

More information

A correlation coefficient for circular data

A correlation coefficient for circular data BiomelriL-a (1983). 70. 2, pp. 327-32 327 Prinltd in Great Britain A correlation coefficient for circular data BY N. I. FISHER CSIRO Division of Mathematics and Statistics, Lindfield, N.S.W., Australia

More information

Testing Goodness-of-Fit for Exponential Distribution Based on Cumulative Residual Entropy

Testing Goodness-of-Fit for Exponential Distribution Based on Cumulative Residual Entropy This article was downloaded by: [Ferdowsi University] On: 16 April 212, At: 4:53 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 172954 Registered office: Mortimer

More information

Sufficiency and conditionality

Sufficiency and conditionality Biometrika (1975), 62, 2, p. 251 251 Printed in Great Britain Sufficiency and conditionality BY JOHN D. KALBFLEISCH Department of Statistics, University of Waterloo, Ontario SUMMARY Ancillary statistics

More information

Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution

Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution Journal of Computational and Applied Mathematics 216 (2008) 545 553 www.elsevier.com/locate/cam Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution

More information

Foundations of Probability and Statistics

Foundations of Probability and Statistics Foundations of Probability and Statistics William C. Rinaman Le Moyne College Syracuse, New York Saunders College Publishing Harcourt Brace College Publishers Fort Worth Philadelphia San Diego New York

More information

simple if it completely specifies the density of x

simple if it completely specifies the density of x 3. Hypothesis Testing Pure significance tests Data x = (x 1,..., x n ) from f(x, θ) Hypothesis H 0 : restricts f(x, θ) Are the data consistent with H 0? H 0 is called the null hypothesis simple if it completely

More information

Semiparametric Regression

Semiparametric Regression Semiparametric Regression Patrick Breheny October 22 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/23 Introduction Over the past few weeks, we ve introduced a variety of regression models under

More information

Dr. Junchao Xia Center of Biophysics and Computational Biology. Fall /1/2016 1/46

Dr. Junchao Xia Center of Biophysics and Computational Biology. Fall /1/2016 1/46 BIO5312 Biostatistics Lecture 10:Regression and Correlation Methods Dr. Junchao Xia Center of Biophysics and Computational Biology Fall 2016 11/1/2016 1/46 Outline In this lecture, we will discuss topics

More information

Recall that in order to prove Theorem 8.8, we argued that under certain regularity conditions, the following facts are true under H 0 : 1 n

Recall that in order to prove Theorem 8.8, we argued that under certain regularity conditions, the following facts are true under H 0 : 1 n Chapter 9 Hypothesis Testing 9.1 Wald, Rao, and Likelihood Ratio Tests Suppose we wish to test H 0 : θ = θ 0 against H 1 : θ θ 0. The likelihood-based results of Chapter 8 give rise to several possible

More information

BIOL 4605/7220 CH 20.1 Correlation

BIOL 4605/7220 CH 20.1 Correlation BIOL 4605/70 CH 0. Correlation GPT Lectures Cailin Xu November 9, 0 GLM: correlation Regression ANOVA Only one dependent variable GLM ANCOVA Multivariate analysis Multiple dependent variables (Correlation)

More information

ON LARGE SAMPLE PROPERTIES OF CERTAIN NONPARAMETRIC PROCEDURES

ON LARGE SAMPLE PROPERTIES OF CERTAIN NONPARAMETRIC PROCEDURES ON LARGE SAMPLE PROPERTIES OF CERTAIN NONPARAMETRIC PROCEDURES 1. Summary and introduction HERMAN RUBIN PURDUE UNIVERSITY Efficiencies of one sided and two sided procedures are considered from the standpoint

More information

Math 423/533: The Main Theoretical Topics

Math 423/533: The Main Theoretical Topics Math 423/533: The Main Theoretical Topics Notation sample size n, data index i number of predictors, p (p = 2 for simple linear regression) y i : response for individual i x i = (x i1,..., x ip ) (1 p)

More information

Subject CS1 Actuarial Statistics 1 Core Principles

Subject CS1 Actuarial Statistics 1 Core Principles Institute of Actuaries of India Subject CS1 Actuarial Statistics 1 Core Principles For 2019 Examinations Aim The aim of the Actuarial Statistics 1 subject is to provide a grounding in mathematical and

More information

Bayesian inference for sample surveys. Roderick Little Module 2: Bayesian models for simple random samples

Bayesian inference for sample surveys. Roderick Little Module 2: Bayesian models for simple random samples Bayesian inference for sample surveys Roderick Little Module : Bayesian models for simple random samples Superpopulation Modeling: Estimating parameters Various principles: least squares, method of moments,

More information

PSY 307 Statistics for the Behavioral Sciences. Chapter 20 Tests for Ranked Data, Choosing Statistical Tests

PSY 307 Statistics for the Behavioral Sciences. Chapter 20 Tests for Ranked Data, Choosing Statistical Tests PSY 307 Statistics for the Behavioral Sciences Chapter 20 Tests for Ranked Data, Choosing Statistical Tests What To Do with Non-normal Distributions Tranformations (pg 382): The shape of the distribution

More information

Preface Introduction to Statistics and Data Analysis Overview: Statistical Inference, Samples, Populations, and Experimental Design The Role of

Preface Introduction to Statistics and Data Analysis Overview: Statistical Inference, Samples, Populations, and Experimental Design The Role of Preface Introduction to Statistics and Data Analysis Overview: Statistical Inference, Samples, Populations, and Experimental Design The Role of Probability Sampling Procedures Collection of Data Measures

More information

Detection of Influential Observation in Linear Regression. R. Dennis Cook. Technometrics, Vol. 19, No. 1. (Feb., 1977), pp

Detection of Influential Observation in Linear Regression. R. Dennis Cook. Technometrics, Vol. 19, No. 1. (Feb., 1977), pp Detection of Influential Observation in Linear Regression R. Dennis Cook Technometrics, Vol. 19, No. 1. (Feb., 1977), pp. 15-18. Stable URL: http://links.jstor.org/sici?sici=0040-1706%28197702%2919%3a1%3c15%3adoioil%3e2.0.co%3b2-8

More information

Solutions to the Spring 2015 CAS Exam ST

Solutions to the Spring 2015 CAS Exam ST Solutions to the Spring 2015 CAS Exam ST (updated to include the CAS Final Answer Key of July 15) There were 25 questions in total, of equal value, on this 2.5 hour exam. There was a 10 minute reading

More information

Step-Stress Models and Associated Inference

Step-Stress Models and Associated Inference Department of Mathematics & Statistics Indian Institute of Technology Kanpur August 19, 2014 Outline Accelerated Life Test 1 Accelerated Life Test 2 3 4 5 6 7 Outline Accelerated Life Test 1 Accelerated

More information

Fiducial Inference and Generalizations

Fiducial Inference and Generalizations Fiducial Inference and Generalizations Jan Hannig Department of Statistics and Operations Research The University of North Carolina at Chapel Hill Hari Iyer Department of Statistics, Colorado State University

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Queues with Time-Dependent Arrival Rates: II. The Maximum Queue and the Return to Equilibrium Author(s): G. F. Newell Source: Journal of Applied Probability, Vol. 5, No. 3 (Dec., 1968), pp. 579-590 Published

More information

Interval Estimation for Parameters of a Bivariate Time Varying Covariate Model

Interval Estimation for Parameters of a Bivariate Time Varying Covariate Model Pertanika J. Sci. & Technol. 17 (2): 313 323 (2009) ISSN: 0128-7680 Universiti Putra Malaysia Press Interval Estimation for Parameters of a Bivariate Time Varying Covariate Model Jayanthi Arasan Department

More information

Theory and Methods of Statistical Inference. PART I Frequentist theory and methods

Theory and Methods of Statistical Inference. PART I Frequentist theory and methods PhD School in Statistics cycle XXVI, 2011 Theory and Methods of Statistical Inference PART I Frequentist theory and methods (A. Salvan, N. Sartori, L. Pace) Syllabus Some prerequisites: Empirical distribution

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Biometrika Trust The Use of a Concomitant Variable in Selecting an Experimental Design Author(s): D. R. Cox Source: Biometrika, Vol. 44, No. 1/2 (Jun., 1957), pp. 150-158 Published by: Oxford University

More information

Kumaun University Nainital

Kumaun University Nainital Kumaun University Nainital Department of Statistics B. Sc. Semester system course structure: 1. The course work shall be divided into six semesters with three papers in each semester. 2. Each paper in

More information

HYPOTHESIS TESTING II TESTS ON MEANS. Sorana D. Bolboacă

HYPOTHESIS TESTING II TESTS ON MEANS. Sorana D. Bolboacă HYPOTHESIS TESTING II TESTS ON MEANS Sorana D. Bolboacă OBJECTIVES Significance value vs p value Parametric vs non parametric tests Tests on means: 1 Dec 14 2 SIGNIFICANCE LEVEL VS. p VALUE Materials and

More information

Sequential Procedure for Testing Hypothesis about Mean of Latent Gaussian Process

Sequential Procedure for Testing Hypothesis about Mean of Latent Gaussian Process Applied Mathematical Sciences, Vol. 4, 2010, no. 62, 3083-3093 Sequential Procedure for Testing Hypothesis about Mean of Latent Gaussian Process Julia Bondarenko Helmut-Schmidt University Hamburg University

More information

Deccan Education Society s FERGUSSON COLLEGE, PUNE (AUTONOMOUS) SYLLABUS UNDER AUTOMONY. SECOND YEAR B.Sc. SEMESTER - III

Deccan Education Society s FERGUSSON COLLEGE, PUNE (AUTONOMOUS) SYLLABUS UNDER AUTOMONY. SECOND YEAR B.Sc. SEMESTER - III Deccan Education Society s FERGUSSON COLLEGE, PUNE (AUTONOMOUS) SYLLABUS UNDER AUTOMONY SECOND YEAR B.Sc. SEMESTER - III SYLLABUS FOR S. Y. B. Sc. STATISTICS Academic Year 07-8 S.Y. B.Sc. (Statistics)

More information

Robust factorial ANCOVA with LTS error distributions

Robust factorial ANCOVA with LTS error distributions Hacettepe Journal of Mathematics and Statistics Volume 47 (2) (2018), 347 363 Robust factorial ANCOVA with LTS error distributions Şükrü Acıtaş and Birdal Şenoğlu Abstract In this study, parameter estimation

More information

Theory and Methods of Statistical Inference

Theory and Methods of Statistical Inference PhD School in Statistics cycle XXIX, 2014 Theory and Methods of Statistical Inference Instructors: B. Liseo, L. Pace, A. Salvan (course coordinator), N. Sartori, A. Tancredi, L. Ventura Syllabus Some prerequisites:

More information

PENALIZED LIKELIHOOD PARAMETER ESTIMATION FOR ADDITIVE HAZARD MODELS WITH INTERVAL CENSORED DATA

PENALIZED LIKELIHOOD PARAMETER ESTIMATION FOR ADDITIVE HAZARD MODELS WITH INTERVAL CENSORED DATA PENALIZED LIKELIHOOD PARAMETER ESTIMATION FOR ADDITIVE HAZARD MODELS WITH INTERVAL CENSORED DATA Kasun Rathnayake ; A/Prof Jun Ma Department of Statistics Faculty of Science and Engineering Macquarie University

More information

STAT331. Cox s Proportional Hazards Model

STAT331. Cox s Proportional Hazards Model STAT331 Cox s Proportional Hazards Model In this unit we introduce Cox s proportional hazards (Cox s PH) model, give a heuristic development of the partial likelihood function, and discuss adaptations

More information

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at The Analysis of Multivariate Binary Data Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 21, No. 2 (1972), pp. 113-120 Published by: Wiley for

More information

Practice Problems Section Problems

Practice Problems Section Problems Practice Problems Section 4-4-3 4-4 4-5 4-6 4-7 4-8 4-10 Supplemental Problems 4-1 to 4-9 4-13, 14, 15, 17, 19, 0 4-3, 34, 36, 38 4-47, 49, 5, 54, 55 4-59, 60, 63 4-66, 68, 69, 70, 74 4-79, 81, 84 4-85,

More information

Probability Distributions.

Probability Distributions. Probability Distributions http://www.pelagicos.net/classes_biometry_fa18.htm Probability Measuring Discrete Outcomes Plotting probabilities for discrete outcomes: 0.6 0.5 0.4 0.3 0.2 0.1 NOTE: Area within

More information

Cornell University and Colorado State University

Cornell University and Colorado State University \ PERMUTATION MOMENTS OF A MULTIVARIATE TEST STATISTIC by N, S. Urquhart and J S. Williams Cornell University and Colorado State University ABSTRACT Many tests for mean differences utilize statistics based

More information

Distribution-Free Tests for Two-Sample Location Problems Based on Subsamples

Distribution-Free Tests for Two-Sample Location Problems Based on Subsamples 3 Journal of Advanced Statistics Vol. No. March 6 https://dx.doi.org/.66/jas.6.4 Distribution-Free Tests for Two-Sample Location Problems Based on Subsamples Deepa R. Acharya and Parameshwar V. Pandit

More information

OPTIMUM DESIGN ON STEP-STRESS LIFE TESTING

OPTIMUM DESIGN ON STEP-STRESS LIFE TESTING Libraries Conference on Applied Statistics in Agriculture 1998-10th Annual Conference Proceedings OPTIMUM DESIGN ON STEP-STRESS LIFE TESTING C. Xiong Follow this and additional works at: http://newprairiepress.org/agstatconference

More information

Introduction to Reliability Theory (part 2)

Introduction to Reliability Theory (part 2) Introduction to Reliability Theory (part 2) Frank Coolen UTOPIAE Training School II, Durham University 3 July 2018 (UTOPIAE) Introduction to Reliability Theory 1 / 21 Outline Statistical issues Software

More information

Inferences on a Normal Covariance Matrix and Generalized Variance with Monotone Missing Data

Inferences on a Normal Covariance Matrix and Generalized Variance with Monotone Missing Data Journal of Multivariate Analysis 78, 6282 (2001) doi:10.1006jmva.2000.1939, available online at http:www.idealibrary.com on Inferences on a Normal Covariance Matrix and Generalized Variance with Monotone

More information

A Monte-Carlo study of asymptotically robust tests for correlation coefficients

A Monte-Carlo study of asymptotically robust tests for correlation coefficients Biometrika (1973), 6, 3, p. 661 551 Printed in Great Britain A Monte-Carlo study of asymptotically robust tests for correlation coefficients BY G. T. DUNCAN AND M. W. J. LAYAKD University of California,

More information

Chapter 1 Statistical Inference

Chapter 1 Statistical Inference Chapter 1 Statistical Inference causal inference To infer causality, you need a randomized experiment (or a huge observational study and lots of outside information). inference to populations Generalizations

More information

A comparison study of the nonparametric tests based on the empirical distributions

A comparison study of the nonparametric tests based on the empirical distributions 통계연구 (2015), 제 20 권제 3 호, 1-12 A comparison study of the nonparametric tests based on the empirical distributions Hyo-Il Park 1) Abstract In this study, we propose a nonparametric test based on the empirical

More information

Table of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z).

Table of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z). Table of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z). For example P(X.04) =.8508. For z < 0 subtract the value from,

More information

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE THE ROYAL STATISTICAL SOCIETY 004 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER II STATISTICAL METHODS The Society provides these solutions to assist candidates preparing for the examinations in future

More information

Math 562 Homework 1 August 29, 2006 Dr. Ron Sahoo

Math 562 Homework 1 August 29, 2006 Dr. Ron Sahoo Math 56 Homework August 9, 006 Dr. Ron Sahoo He who labors diligently need never despair; for all things are accomplished by diligence and labor. Menander of Athens Direction: This homework worths 60 points

More information

EVALUATING THE REPEATABILITY OF TWO STUDIES OF A LARGE NUMBER OF OBJECTS: MODIFIED KENDALL RANK-ORDER ASSOCIATION TEST

EVALUATING THE REPEATABILITY OF TWO STUDIES OF A LARGE NUMBER OF OBJECTS: MODIFIED KENDALL RANK-ORDER ASSOCIATION TEST EVALUATING THE REPEATABILITY OF TWO STUDIES OF A LARGE NUMBER OF OBJECTS: MODIFIED KENDALL RANK-ORDER ASSOCIATION TEST TIAN ZHENG, SHAW-HWA LO DEPARTMENT OF STATISTICS, COLUMBIA UNIVERSITY Abstract. In

More information

Direction: This test is worth 250 points and each problem worth points. DO ANY SIX

Direction: This test is worth 250 points and each problem worth points. DO ANY SIX Term Test 3 December 5, 2003 Name Math 52 Student Number Direction: This test is worth 250 points and each problem worth 4 points DO ANY SIX PROBLEMS You are required to complete this test within 50 minutes

More information

Theory and Methods of Statistical Inference. PART I Frequentist likelihood methods

Theory and Methods of Statistical Inference. PART I Frequentist likelihood methods PhD School in Statistics XXV cycle, 2010 Theory and Methods of Statistical Inference PART I Frequentist likelihood methods (A. Salvan, N. Sartori, L. Pace) Syllabus Some prerequisites: Empirical distribution

More information

THE PRINCIPLES AND PRACTICE OF STATISTICS IN BIOLOGICAL RESEARCH. Robert R. SOKAL and F. James ROHLF. State University of New York at Stony Brook

THE PRINCIPLES AND PRACTICE OF STATISTICS IN BIOLOGICAL RESEARCH. Robert R. SOKAL and F. James ROHLF. State University of New York at Stony Brook BIOMETRY THE PRINCIPLES AND PRACTICE OF STATISTICS IN BIOLOGICAL RESEARCH THIRD E D I T I O N Robert R. SOKAL and F. James ROHLF State University of New York at Stony Brook W. H. FREEMAN AND COMPANY New

More information

14.30 Introduction to Statistical Methods in Economics Spring 2009

14.30 Introduction to Statistical Methods in Economics Spring 2009 MIT OpenCourseWare http://ocw.mit.edu 4.0 Introduction to Statistical Methods in Economics Spring 009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Simulation-based robust IV inference for lifetime data

Simulation-based robust IV inference for lifetime data Simulation-based robust IV inference for lifetime data Anand Acharya 1 Lynda Khalaf 1 Marcel Voia 1 Myra Yazbeck 2 David Wensley 3 1 Department of Economics Carleton University 2 Department of Economics

More information

Formulas and Tables for Elementary Statistics, Eighth Edition, by Mario F. Triola 2001 by Addison Wesley Longman Publishing Company, Inc.

Formulas and Tables for Elementary Statistics, Eighth Edition, by Mario F. Triola 2001 by Addison Wesley Longman Publishing Company, Inc. Formulas and Tables for Elementary Statistics, Eighth Edition, by Mario F. Triola 2001 by Addison Wesley Longman Publishing Company, Inc. Ch. 2: Descriptive Statistics x Sf. x x Sf Mean S(x 2 x) 2 s 2

More information

On robust and efficient estimation of the center of. Symmetry.

On robust and efficient estimation of the center of. Symmetry. On robust and efficient estimation of the center of symmetry Howard D. Bondell Department of Statistics, North Carolina State University Raleigh, NC 27695-8203, U.S.A (email: bondell@stat.ncsu.edu) Abstract

More information

Minimum distance tests and estimates based on ranks

Minimum distance tests and estimates based on ranks Minimum distance tests and estimates based on ranks Authors: Radim Navrátil Department of Mathematics and Statistics, Masaryk University Brno, Czech Republic (navratil@math.muni.cz) Abstract: It is well

More information

1 One-way Analysis of Variance

1 One-way Analysis of Variance 1 One-way Analysis of Variance Suppose that a random sample of q individuals receives treatment T i, i = 1,,... p. Let Y ij be the response from the jth individual to be treated with the ith treatment

More information

Introduction to Statistical Analysis

Introduction to Statistical Analysis Introduction to Statistical Analysis Changyu Shen Richard A. and Susan F. Smith Center for Outcomes Research in Cardiology Beth Israel Deaconess Medical Center Harvard Medical School Objectives Descriptive

More information