The Relative Efficiency of Goodness-of-Fit Statistics in the Simple and Composite Hypothesis-Testing Problem
本文比较了简单和复合假设下Cramer-von Mises型统计量的检验效率，利用Bahadur效率和Pitman效率解释了Dyer和Stephens模拟中参数估计使检验更有效的悖论。
Abstract Let X 1, …., X n be a sequence of independent and identically distributed random variables with an unknown underlying continuous cumulative distribution function F. Relative to this unknown distribution function, suppose one would like to test a null hypothesis concerning the goodness of fit of F to some distribution function using a Cramer-von Mises-type statistic. In some applications the null hypothesis is simple, whereas in others it may be composite. The subject of this article is the problem of comparing the efficiency of tests for the simple and composite hypotheses. Dyer (1974) and Stephens (1974) noticed that for testing normality using the Cramer-von Mises statistic the test procedures for which the nuisance parameters needed to be estimated were more powerful than the procedures in which the parameters were specified. We shall analytically compare these tests of the simple and composite hypotheses using the concepts of Bahadur efficiency and Pitman efficiency and thereby give a theoretical basis for the paradoxical simulation results of Dyer and Stephens. Key Words: Approximate Bahadur efficiencyCramer-von Mises-type statisticEigenvaluesGoodness-of-fit testsPitman efficiency