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

Abstract

Suppose that at any stage of a statistical experiment a control variable XX that affects the distribution of the observed data YY can be used. The distribution of YY depends on some unknown parameter θ\theta, and we consider the classical problem of testing a simple hypothesis H0:θ=θ0H_0: \theta=\theta_0 against a simple alternative H1:θ=θ1H_1: \theta=\theta_1 allowing the data to be controlled by XX, in the following sequential context. The experiment starts with assigning a value X1X_1 to the control variable and observing Y1Y_1 as a response. After some analysis, we choose another value X2X_2 for the control variable, and observe Y2Y_2 as a response, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favour of H0H_0 or H1H_1 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.

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