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Distribution theory of nonparametric test statistics and sum of generalized Lindley random variables

木谷 聖人 Masato Kitani 東京理科大学 DOI:info:doi/10.20604/00003672

2022.06.17

概要

Introduction
One-sample and two-sample testing problems are important topics in statistics. Samples are com- pared using conventional tests based on the assumption that a single population or a population of differences between pairs is normal. Researchers commonly assume a normal distribution when they analyse the experimental data. However, the assumption of normality is often inappropriate in practice. As B¨uning (1997) and Nanna and Sawilowsky (1998) pointed out, normality is the exception rather than the rule. Micceri (1989) investigated 440 large research data sets in psy- chology. Regarding symmetry and tails, less than 7% of these data sets were similar to a normal distribution. As a matter of fact, every data set was non-normal at the 1% significance level. Hence, the nonparametric procedure is required when normality cannot clearly be assumed for a specific distribution. In the 1940s, the rank-based approach emerged. Initiated by Wilcoxon (1945), vari- ous nonparametric tests were subsequently developed by Mann and Whitney (1947), Mood (1954), Ansari and Bradley (1960), and many others. In the 1950s and 1960s, Pitman (1948), Hodges and Lehmann (1956), and Chernoff and Savage (1958) showed that nonparametric tests have desirable efficiency properties relative to their parametric counterparts. Their work has contributed to the use of nonparametric methods in experimental design and regression analysis. Several statistical resampling methods, such as the jackknife and the bootstrap introduced by Efron (1979), devel- oped since the 1980s, make use of the computational power of computers to provide standard errors and confidence intervals in many applications, including complicated ones where it is difficult, if not impossible, to use a parametric approach. Nowadays, with enhanced computer performance, nonparametric methods have become the mainstream analysis method.

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