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Optimal sequential multiple hypothesis testing in presence of control variables. (English) Zbl 1165.62053
Summary: 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 \(\theta \), and we consider the problem of testing the multiple hypotheses \(H_{1}: \theta =\theta _{1}, H_{2}: \theta =\theta _{2}, \dots , H_{k}: \theta =\theta _{k}\) allowing the data to be controlled by X, in the following sequential context.
The experiment starts with assigning a value \(X_{1}\) to the control variable and observing \(Y_{1}\) as a response. After some analysis, another value \(X_{2}\) for the control variable is chosen, and \(Y_{2}\) 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 \(H_{1}, \dots, H_{k}\) is to be taken. 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 \(Y_{1}, Y_{2}, \dots , Y_{n}\) are independent, given the controls \(X_{1}, X_{2}, \dots , X_{n}\), n=1,2,\(\dots \).

MSC:
62L10 Sequential statistical analysis
62L15 Optimal stopping in statistics
93E20 Optimal stochastic control
62C99 Statistical decision theory
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