Testing one Simple Hypothesis Against Another

Abstract

For the problem of testing one simple hypothesis against another, of all tests whose probabilities of incorrectly accepting the first hypothesis and of incorrectly accepting the second hypothesis are bounded from above by given bounds, the familiar Wald sequential probability ratio test gives the smallest expectation of sample size under either hypothesis. In this paper, a "generalized sequential probability ratio test" is introduced which differs from the Wald test only in that the same limits ($A, B$ in the usual notation) are not necessarily used at each stage of the sampling, but at the $i$th stage $A_i$ and $B_i$ are used, where these numbers are predetermined constants. It is shown that for any given test $T$, there is a generalized sequential probability ratio test $G$ whose probabilities of incorrectly accepting either hypothesis are no larger than the corresponding probabilities for $T$, and such that the cumulative distribution function of the number of observations required to come to a decision when using $G$ is never below the corresponding distribution function when using $T$, under either hypothesis. We may then say that "$G$ is uniformly better than $T$."