Design considerations for neyman-pearson and wald hypothesis testing

Abstract

The Neyman-Pearson Lemma describes a test for two simple hypotheses that, for a given sample size, is most powerful for its level. It is usually implemented by choosing the smallest sample size that achieves a prespecified power for a fixed level. The Lemma does not describe how to select either the level or the power of the test. In the usual Wald decision-theoretic structure there exists a sampling cost function, an initial prior over the hypothesis space and various payoffs to right/wrong hypothesis selections. The optimal Wald test is a Bayes decision rule that maximizes the expected payoff net of sampling costs. This paper shows that the Wald-optimal test and the Neyman-Pearson test can be the same and how the Neyman-Pearson test, with fixed level and power, can be viewed as a Wald test subject to restrictions on the payoff vector, cost function and prior distribution.