Stability Estimating in Optimal Sequential Hypotheses Testing

We study the stability of the classical optimal sequential probability ratio test based on independent identically distributed observations X1, X2, . . . when testing two simple hypotheses about their common density f : f = f0 versus f = f1. As a functional to be minimized, it is used a weighted sum of the average (under f0) sample number and the two types error probabilities. We prove that the problem is reduced to stopping time optimization for a ratio process generated by X1, X2, . . . with the density f0. For τ∗ being the corresponding optimal stopping time we consider a situation when this rule is applied for testing between f0 and an alternative f̃1, where f̃1 is some approximation to f1. An inequality is obtained which gives an upper bound for the expected cost excess, when τ∗ is used instead of the rule τ̃∗ optimal for the pair (f0, f̃1). The inequality found also estimates the difference between the minimal expected costs for optimal tests corresponding to the pairs (f0, f1) and (f0, f̃1).