Date of Award
Doctor of Philosophy
Let X1, ...,Xm and Y1, ...,Yn be two independent samples from two continuous distributions F and G. For testing the hypothesis H0: F = G, with no additional assumption on F and G, rank tests are used as they are easy to use and distribution-free under H0. However, the problem of finding the distribution of the ranks of the Xs and Ys in the pooled sample is difficult in general when F ≠ G. Lehmann (1953) specialized a theorem of Hoeffding (1951, pages 87-88), and used it to obtain the distribution of the ranks under the alternatives G = Q(F), where Q is an absolutely continuous distribution function on (0,1). However, the specialization cannot be used directly, even for the location shift problem, when the supports of the two distributions are not the same and no one of the supports contains the other. A generalized Hoeffding type theorem is given and used to obtain the distribution of the ranks under a variety of alternatives. Hence the small sample powers of the most frequently used rank tests for two-sample location and scale problems are obtained;The best precedence test (BPT) is derived for testing H0 in a life testing experiment when two types of items are on test. The test has maximum power in the class of precedence tests at a given alternative F = 1 - (1 - G)[superscript][lambda] for some [lambda] > 1. The additional advantage in using the BPT is that it saves considerable time on test. We compare the power of the BPT with other tests and also obtain the average number of failures for the BPT;Consider the class of truncated populations such that G (F) is a left (right or left-and-right) truncation of F (G), or F and G are different types of truncation of some other distribution H. This class includes location-shift exponential distributions, location-shift uniform distributions and scale-change uniform distributions. The distribution of the ranks under each of the truncation models is obtained in a simple form. The ordering of the values of the probability function of the ranks is used to search for locally most powerful tests;When the observations are censored using different censoring models, e.g., censoring to the right at a fixed point, random censoring to the right, etc., the distribution of the ranks has been studied. Hence the powers of the generalized Wilcoxon test and the logrank test under the random censoring model are obtained;References: Hoeffding, W. (1951). Optimum Nonparametric Tests. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 83-92. Lehmann, E. L. (1953). The Power of Rank Tests. Annals of Mathematical Statistics, 24, 23-43.
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Lin, Chiou-Hua, "Powers of two-sample rank tests under Lehmann alternatives " (1990). Retrospective Theses and Dissertations. 11200.