More Optimality Properties of the Sequential Probability Ratio Test

Consider the problem of testing sequentially the null hypothesis “ $$\theta = 0$$ ” against the alternative “ $$ \theta = 1 $$ ” on the basis of i.i.d. potentially observable variables X1, X2,…. Let N be a stopping rule admitting a test based on (Xi,…, XN) having probabilities of errors ao and ai. Then the Hellinger transform of (Xi,…, XN) is at most equal to that of (X1,..., XN*.) where N* is the stopping rule of a sequential probability ratio test 5‘ having the same probabilities of errors. In particular the Hellinger distance between the distributions of (X1,…, XN) under $$ \theta = 0 $$ and $$ \theta = 1 $$ is at least equal to the same distance for (X1,…, XN*.). This remains so if the Hellinger distance is replaced by the statistical distance and provided the number 1 is not outside the stopping bounds.