Uniformly Most Powerful Tests
Prove the Sign Test is a most powerful testand does
Sungsub Choi , W. Tests of hypotheses about finite-dimensional parameters in a semiparametric model are studied from Pitman's moving alternative or local approach using Le Cam's local asymptotic normality concept. For the case of a real parameter being tested, asymptotically uniformly most powerful AUMP tests are characterized for one-sided hypotheses, and AUMP unbiased tests for two-sided ones. An asymptotic invariance principle is introduced for multidimensional hypotheses, and AUMP invariant tests are characterized. These provide optimality for Wald, Rao score , Neyman-Rao effective score and likelihood ratio tests in parametric models, and for Neyman-Rao tests in semiparametric models when constructions are feasible. Inversions lead to asymptotically uniformly most accurate confidence sets. Examples include one-, two- and k -sample problems, a linear regression model with unknown error distribution and a proportional hazards regression model with arbitrary baseline hazards.
Simple versus Simple testing Theorem : For each fixed the quantity is minimized by any which has. Neyman and Pearson suggested that in practice the two kinds of errors might well have unequal consequences. They suggested that rather than minimize any quantity of the form above you pick the more serious kind of error, label it Type I and require your rule to hold the probability of a Type I error to be no more than some prespecified level. This value is typically 0. The Neyman and Pearson approach is then to minimize subject to the constraint.
Assume an independently and identically distributed i. Figure 2. Observed realizations from this i. The power function is an important tool for accessing the quality of a test and for comparing different test procedures. Conservative Test: If possible, a test is constructed in such a way that size equals level, i. In this case the test is called conservative. Most Powerful Test: When choosing between different testing procedures for the same testing problem, one will usually prefer the most powerful test.
Uniformly most powerful tests are statistical hypothesis tests that provide the greatest power against a fixed null hypothesis among all tests of a given size. In this article, the notion of uniformly most powerful tests is extended to the Bayesian setting by defining uniformly most powerful Bayesian tests to be tests that maximize the probability that the Bayes factor, in favor of the alternative hypothesis, exceeds a specified threshold. Like their classical counterpart, uniformly most powerful Bayesian tests are most easily defined in one-parameter exponential family models, although extensions outside of this class are possible.
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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. We close with a brief comparison of our methodology to the Karlin-Rubin theorem.
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A statistical hypothesis test is a method of making statistical decisions from and about experimental data. Null-hypothesis testing just answers the question of "how well the findings fit the possibility that chance factors alone might be responsible. One use is deciding whether experimental results contain enough information to cast doubt on conventional wisdom.
UNIFORMLY MOST POWERFUL BAYESIAN TESTS
For example, according to the Neyman—Pearson lemma , the likelihood-ratio test is UMP for testing simple point hypotheses. The Karlin—Rubin theorem can be regarded as an extension of the Neyman—Pearson lemma for composite hypotheses. Although the Karlin-Rubin theorem may seem weak because of its restriction to scalar parameter and scalar measurement, it turns out that there exist a host of problems for which the theorem holds. In particular, the one-dimensional exponential family of probability density functions or probability mass functions with. We then have.