# Article

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Keywords:
sequential analysis; sequential hypothesis testing; multiple hypotheses; control variable; independent observations; optimal stopping; optimal control; optimal decision; optimal sequential testing procedure; Bayes; sequential probability ratio test
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 multiple hypotheses $H_1:\,\theta=\theta_1$, $H_2:\,\theta=\theta_2, \ldots$, $H_k:\,\theta=\theta_k$ 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 $H_1,\ldots$, $H_k$ is to be taken. In this article, 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 $Y_1, Y_2,\ldots , Y_n$ are independent, given controls $X_1,X_2,\ldots , X_n$, $n=1,2,\ldots$.
References:
[1] N. Cressie and P. B. Morgan: The VRPT: A sequential testing procedure dominating the SPRT. Econometric Theory 9 (1993), 431–450. MR 1241983
[2] M. Ghosh, N. Mukhopadhyay, and P. K. Sen: Sequential Estimation. John Wiley, New York – Chichester – Weinheim – Brisbane – Singapore – Toronto 1997. MR 1434065
[3] G. W. Haggstrom: Optimal stopping and experimental design. Ann. Math. Statist. 37 (1966), 7–29. MR 0195221 | Zbl 0202.49201
[4] G. Lorden: Structure of sequential tests minimizing an expected sample size. Z. Wahrsch. verw. Geb. 51 (1980), 291–302. MR 0566323 | Zbl 0407.62055
[5] M. B. Malyutov: Lower bounds for the mean length of a sequentially planned experiment. Soviet Math. (Iz. VUZ) 27 (1983), 11, 21–47. MR 0733570
[6] A. Novikov: Optimal sequential testing of two simple hypotheses in presence of control variables. Internat. Math. Forum 3 (2008), 41, 2025–2048. Preprint arXiv:0812.1395v1 [math.ST] ( http://arxiv.org/abs/0812.1395) MR 2470661
[7] A. Novikov: Optimal sequential multiple hypothesis tests. Kybernetika 45 (2009), 2, 309–330. MR 2518154 | Zbl 1167.62453
[8] A. Novikov: Optimal sequential procedures with Bayes decision rules. Preprint arXiv:0812.0159v1 [math.ST]( http://arxiv.org/abs/0812.0159) MR 2685120
[9] A. Novikov: Optimal sequential tests for two simple hypotheses based on independent observations. Internat. J. Pure Appl. Math. 45 (2008), 2, 291–314. MR 2421867
[10] N. Schmitz: Optimal Sequentially Planned Decision Procedures. (Lecture Notes in Statistics 79.) Springer-Verlag, New York 1993. MR 1226454 | Zbl 0771.62057
[11] I. N. Volodin: Guaranteed statistical inference procedures (determination of the optimal sample size). J. Math. Sci. 44 (1989), 5, 568–600. MR 0885413 | Zbl 0666.62077
[12] A. Wald and J. Wolfowitz: Optimum character of the sequential probability ratio test. Ann. Math. Statist. 19 (1948), 326–339. MR 0026779
[13] S. Zacks: The Theory of Statistical Inference. John Wiley, New York – London – Sydney – Toronto 1971. MR 0420923 | Zbl 0321.62003

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