Optimal sequential testing of two simple hypotheses in presence of control variables

Suppose that at any stage of a statistical experiment a control variable that affects the distribution of the observed data can be used. The distribution of depends on some unknown parameter , and we consider the classical problem of testing a simple hypothesis against a simple alternative allowing the data to be controlled by , in the following sequential context. The experiment starts with assigning a value to the control variable and observing as a response. After some analysis, we choose another value for the control variable, and observe as a response, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favour of or is to be taken. In this article, our aim is to characterize the structure of optimal sequential procedures, based on this type of data, for testing a simple hypothesis against a simple alternative.
View on arXiv