Optimal sequential multiple hypothesis testing in presence of control variables

Andrey Novikov

Kybernetika (2009)

  • Volume: 45, Issue: 3, page 507-528
  • ISSN: 0023-5954

Abstract

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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 θ , and we consider the problem of testing multiple hypotheses H 1 : θ = θ 1 , H 2 : θ = θ 2 , ... , H k : θ = θ 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 , ... , 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 , ... , Y n are independent, given controls X 1 , X 2 , ... , X n , n = 1 , 2 , ... .

How to cite

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Novikov, Andrey. "Optimal sequential multiple hypothesis testing in presence of control variables." Kybernetika 45.3 (2009): 507-528. <http://eudml.org/doc/37678>.

@article{Novikov2009,
abstract = {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 $.},
author = {Novikov, Andrey},
journal = {Kybernetika},
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; 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},
language = {eng},
number = {3},
pages = {507-528},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Optimal sequential multiple hypothesis testing in presence of control variables},
url = {http://eudml.org/doc/37678},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Novikov, Andrey
TI - Optimal sequential multiple hypothesis testing in presence of control variables
JO - Kybernetika
PY - 2009
PB - Institute of Information Theory and Automation AS CR
VL - 45
IS - 3
SP - 507
EP - 528
AB - 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 $.
LA - eng
KW - 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; 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
UR - http://eudml.org/doc/37678
ER -

References

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  1. The VRPT: A sequential testing procedure dominating the SPRT, Econometric Theory 9 (1993), 431–450. MR1241983
  2. Sequential Estimation, John Wiley, New York – Chichester – Weinheim – Brisbane – Singapore – Toronto 1997. MR1434065
  3. Optimal stopping and experimental design, Ann. Math. Statist. 37 (1966), 7–29. Zbl0202.49201MR0195221
  4. Structure of sequential tests minimizing an expected sample size, Z. Wahrsch. verw. Geb. 51 (1980), 291–302. Zbl0407.62055MR0566323
  5. Lower bounds for the mean length of a sequentially planned experiment, Soviet Math. (Iz. VUZ) 27 (1983), 11, 21–47. MR0733570
  6. 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) MR2470661
  7. Optimal sequential multiple hypothesis tests, Kybernetika 45 (2009), 2, 309–330. Zbl1167.62453MR2518154
  8. Optimal sequential procedures with Bayes decision rules, Preprint arXiv:0812.0159v1 [math.ST]( http://arxiv.org/abs/0812.0159) MR2685120
  9. Optimal sequential tests for two simple hypotheses based on independent observations, Internat. J. Pure Appl. Math. 45 (2008), 2, 291–314. MR2421867
  10. Optimal Sequentially Planned Decision Procedures, (Lecture Notes in Statistics 79.) Springer-Verlag, New York 1993. Zbl0771.62057MR1226454
  11. Guaranteed statistical inference procedures (determination of the optimal sample size), J. Math. Sci. 44 (1989), 5, 568–600. Zbl0666.62077MR0885413
  12. Optimum character of the sequential probability ratio test, Ann. Math. Statist. 19 (1948), 326–339. MR0026779
  13. The Theory of Statistical Inference, John Wiley, New York – London – Sydney – Toronto 1971. Zbl0321.62003MR0420923

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