Unified Bayesian and conditional frequentist testing procedures

Sarat C Dass, Purdue University

Abstract

In hypothesis testing, the conclusions from Bayesian and Frequentist approaches can differ markedly, especially in the reporting of error probabilities. Recently, Berger, Brown and Wolpert (1994) have shown that the Conditional Frequentist method can be made exactly equivalent to the Bayesian method for simple vs. simple hypothesis testing. This was extended to testing of a simple null versus a composite alternative in Berger, Boukai, and Wang (1997a, 1997b). This thesis extends this unification in two further directions: to composite null hypotheses and to testing in discrete settings. Many composite null hypotheses can be reduced to simple null hypotheses through invariance arguments. The first part of the thesis demonstrates how to construct default Bayesian tests which can similarly be reduced to testing of this simple null hypothesis, allowing the methods of Berger, Boukai, and Wang to be applied. To achieve the unification, one must carefully choose the prior distributions used in the Bayesian analysis. Under the null hypothesis, it is shown that the parameters must be assigned the right Haar measure as a noninformative prior. Priors on the alternative must be compatible with this choice, and also deal with the presence of additional parameters (when the alternative is more complex than the null). When additional parameters are present, ideas from modern Bayesian testing theory, such as 'intrinsic priors from fractional or intrinsic Bayes factors' are utilized in the development. Intrinsic priors arising from using intrinsic Bayes factors in hypotheses testing are shown to be proper under general group invariance conditions. For the unified test, the conditional Type I error is constant over the original null hypothesis, and is equal to the posterior probability of the null. This equivalence is of primary importance in the unification. A number of testing scenarios are studied as illustrations of the methodology. Another issue related to testing is studied, namely, the choice of the sample size to achieve desired conditional goals. Unification for testing problems involving discrete distributions is achieved via randomization. The effect of randomization is minimal for decision making, unlike in unconditional frequentist testing, where the effect can be considerable.

Degree

Ph.D.

Advisors

Berger, Purdue University.

Subject Area

Statistics

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