The WAGR Sequential $t$-Test Reaches a Decision with Probability One

Abstract

The WAGR test is a sequential procedure for testing the null hypothesis that the proportion of a normal population greater than a given constant is $p_0$ (given) against the alternative that it is $p_1$ (given). These are equivalent (after a translation) to hypotheses specifying the value of $\mu/\sigma,$ where $\mu$ and $\sigma^2$ are the mean and the variance of the normal population under test. We prove that, with probability one, a decision is reached when the WAGR test is applied. This fact is of importance in its own right; it also has indirect interest because, unless it were true, the standard Wald inequalities on probabilities of error at the two hypothesis points could not be applied.