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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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