On the Existence of Uniformly Most Powerful Bayesian Tests With Application to Non-Central Chi-Squared Tests
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Uniformly most powerful Bayesian tests (UMPBTs) are an objective class of Bayesian hypothesis tests that can be considered the Bayesian counterpart of classical uniformly most powerful tests. Unfortunately, UMPBTs have only been exposed for application in one parameter exponential family models. The purpose of this article is to describe methodology for deriving UMPBTs for a larger class of tests. Specifically, we introduce sufficient conditions for the existence of UMPBTs and propose a unified approach for their derivation. An important application of our methodology is the extension of UMPBTs to testing whether the non-centrality parameter of a chi-squared distribution is zero. The resulting tests have broad applicability, providing default alternative hypotheses to compute Bayes factors in, for example, Pearson's chi-squared test for goodness-of-fit, tests of independence in contingency tables, and likelihood ratio, score and Wald tests. We close with a brief comparison of our methodology to the Karlin-Rubin theorem.
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