Suppose that at any stage of a statistical experiment a control variable X that affects the distribution of the observed data Y at this stage can be used. The distribution of Y depends on some unknown parameter θ, and we consider the problem of testing multiple hypotheses H1:θ=θ1, H2:θ=θ2,…, Hk:θ=θk allowing the data to be controlled by X, in the following sequential context. The experiment starts with assigning a value X1 to the control variable and observing Y1 as a response. After some analysis, another value X2 for the control variable is chosen, and Y2 as a response is observed, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favor of one of the hypotheses H1,…, Hk is to be taken. In this article, our aim is to characterize the structure of optimal sequential testing procedures based on data obtained from an experiment of this type in the case when the observations Y1,Y2,…,Yn are independent, given controls X1,X2,…,Xn, n=1,2,….
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 f~1, where f~1 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 τ~∗ optimal for the pair (f0,f~1). The inequality found also estimates the difference between the minimal expected costs for optimal tests corresponding to the pairs (f0,f1) and (f0,f~1).