Comparison of the Optimal Properties of the Wald and Neyman-Pearson Criteria

Abstract

Let $H_\theta $ be a hypothesis that the theoretical density of probability is $f(x,\theta )$, where $\theta = (\theta ^1 , \ldots ,\theta ^p )$ is an unknown parameter. The statistical problem of distinguishing between two hypotheses $H_0 $ and $H_\theta $ provided $\theta \to \theta _0 = 0$ is considered. The paper gives asymptotic formulas for the necessary number $n(\alpha ,\beta ;\theta )$ of observations when the Neyman-Pearson test is used (Theorem 1) and for the mean number of necessary observations for the Wald sequential test (Theorem 2). The limit of the ratio of these two numbers depends only on a and $\beta $ (probabilities of the errors of first and second kind) (Theorem 3). The variation of this limit with $\alpha $ and $\beta $ is illustrated by a table at the end of the paper.