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This work deals with a general problem of testing multiple hypotheses about the distribution of a discrete-time stochastic process. Both the Bayesian and the conditional settings are considered. The structure of optimal sequential tests is characterized.

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 multiple hypotheses $H_1:\theta ={\theta }_1$, $H_2:\theta ={\theta }_2,...$, $H_k:\theta ={\theta }_k$ 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...

In this article, a general problem of sequential statistical inference for general discrete-time stochastic processes is considered. The problem is to minimize an average sample number given that Bayesian risk due to incorrect decision does not exceed some given bound. We characterize the form of optimal sequential stopping rules in this problem. In particular, we have a characterization of the form of optimal sequential decision procedures when the Bayesian risk includes both the loss due to incorrect...

We study the stability of the classical optimal sequential probability ratio test based on independent identically distributed observations $X_1,X_2,\cdots$ when testing two simple hypotheses about their common density $f$: $f=f_0$ versus $f=f_1$. As a functional to be minimized, it is used a weighted sum of the average (under $f_0$) 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 $X_1,X_2,\cdots$ with the density $f_0$. For ${\tau }_{*}$ being the corresponding...