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Recent developments towards optimality in multiple hypothesis testing

J. Shaffer

Published 2006 in arXiv: Statistics Theory

ABSTRACT

Abstract: There are many different notions of optimality even in testinga single hypothesis. In the multiple testing area, the number of possibilitiesis very much greater. The paper first will describe multiplicity issues thatarise in tests involving a single parameter, and will describe a new optimalityresult in that context. Although the example given is of minimal practicalimportance, it illustrates the crucial dependence of optimality on the precisespecification of the testing problem. The paper then will discuss the typesof expanded optimality criteria that are being considered when hypothesesinvolve multiple parameters, will note a few new optimality results, and willgive selected theoretical references relevant to optimality considerations underthese expanded criteria. 1. IntroductionThere are many notions of optimality in testing a single hypothesis, and many morein testing multiple hypotheses. In this paper, consideration will be limited to casesin which there are a finite number of individual hypotheses, each of which ascribesa specific value to a single parameter in a parametric model, except for a smallbut important extension: consideration of directional hypothesis-pairs concerningsingle parameters, as described below. Furthermore, only procedures for continuousrandom variables will be considered, since if randomization is ruled out, multipletests can always be improved by taking discreteness of random variables into con-sideration, and these considerations are somewhat peripheral to the main issues tobe addressed.The paper will begin by considering a single hypothesis or directional hypothesis-pair, where some of the optimality issues that arise can be illustrated in a simplesituation. Multiple hypotheses will be treated subsequently. Two previous reviewsof optimal results in multiple testing are Hochberg and Tamhane [28] and Shaffer[58]. The former includes results in confidence interval estimation while the latteris restricted to hypothesis testing.2. Tests involving a single parameterTwo conventional types of hypotheses concerning a single parameter areH : θ ≤ 0 vs. A : θ > 0, (2.1)

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