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On Non-Parametric Testing, the Uniform Behaviour of the t-Test, and Related Problems

Joseph P. Romano
Scandinavian Journal of Statistics
Vol. 31, No. 4 (Dec., 2004), pp. 567-584
Stable URL: http://www.jstor.org/stable/4616851
Page Count: 18
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On Non-Parametric Testing, the Uniform Behaviour of the t-Test, and Related Problems
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Abstract

In this article, we revisit some problems in non-parametric hypothesis testing. First, we extend the classical result of Bahadur & Savage [Ann. Math. Statist. 25 (1956) 1115] to other testing problems, and we answer a conjecture of theirs. Other examples considered are testing whether or not the mean is rational, testing goodness-of-fit, and equivalence testing. Next, we discuss the uniform behaviour of the classical t-test. For most non-parametric models, the Bahadur-Savage result yields that the size of the t-test is one for every sample size. Even if we restrict attention to the family of symmetric distributions supported on a fixed compact set, the t-test is not even uniformly asymptotically level α. However, the convergence of the rejection probability is established uniformly over a large family with a very weak uniform integrability type of condition. Furthermore, under such a restriction, the t-test possesses an asymptotic maximin optimality property.

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