Optimum Character of the Sequential Probability Ratio Test

Abstract

Let S0 be any sequential probability ratio test for deciding between two simple alternatives H0 and H1, and S1 another test for the same purpose. We define (i,j=0,1): αi(Sj)= probability, under Sj, of rejecting Hi when it is true; Eji(n)= expected number of observations to reach a decision under test Sj when the hypothesis Hi is true. (It is assumed that E1i(n) exists.) In this paper it is proved that, if αi(S1)≤αi(S0)(i=0,1), it follows that E0i(n)≤E1i(n)(i=0,1). This means that of all tests with the same power the sequential probability ratio test requires on the average fewest observations. This result had been conjectured earlier ([1], [2]).