Exponential rates for the error probabilities in selection procedures. (English) Zbl 1274.62158

Summary: For a sequence of statistical experiments with a finite parameter set the asymptotic behavior of the maximum risk is studied for the problem of classification into disjoint subsets. The exponential rate of the optimal decision rule is determined and expressed in terms of the normalized limit of moment generating functions of likelihood ratios. Necessary and sufficient conditions for the existence of adaptive classification rules in the sense of A. L. Rukhin [Adaptive procedure for a finite numbers of probability distributions. Statistical decision theory and related topics III, Proc. 3rd Purdue Symp., West Lafayette/Indiana 1981, Vol. 2, 269–285 (1982; Zbl 0585.62019)] are given. The results are applied to the problem of the selection of the best population. Exponential families are studied as a special case, and an example for the normal case is included.


62F07 Statistical ranking and selection procedures
62H30 Classification and discrimination; cluster analysis (statistical aspects)


Zbl 0585.62019
Full Text: Link


[1] Bucklew I. A.: Large Deviation Techniques in Decision, Simulation and Estimatio. · Zbl 0665.94006
[2] Chernoff H.: A measure of asymptotic efficiency for tests of hypothesis based on the sum of observation. · Zbl 0048.11804 · doi:10.1214/aoms/1177729330
[3] Chernoff H.: Large sample theory: Parametric cas. · Zbl 0072.35703 · doi:10.1214/aoms/1177728347
[4] Krafft O., Plachky D.: Bounds for the power of likelihood ratio tests and their asymptotic and their asymptotic propertie. · Zbl 0214.18003 · doi:10.1214/aoms/1177696808
[5] Krafft O., Puri M. L.: The asymptotic behaviour of the minimax risk for multiple decision problem. · Zbl 0296.62007
[6] Liese F., Miescke K. L.: Exponential Rates for the Error Probabilities in Selection Procedures. Preprint 96/5, FB Mathematik, Universität Rostock, Rostock 1996
[7] Liese F., Vajda I.: Convex Statistical Distance.
[8] Rüschendorf L.: Asymptotische Statisti.
[9] Rukhin A. L.: Adaptive procedure for a finite numbers of probability distributions. Statist. Decis. Theory Related Topics III. 2 (1982), 269-285
[10] Rukhin A. L.: Adaptive classification procedure. · Zbl 0572.62050 · doi:10.2307/2288284
[11] Rukhin A. L.: Adaptive testing of multiple hypotheses for stochastic processe. · Zbl 0804.62077 · doi:10.1080/07362999408809374
[12] Rukhin A. L., Vajda I.: Adaptive decision making for stochastic processe. · Zbl 0829.62095 · doi:10.1016/0378-3758(93)00087-X
[13] Vajda I.: Theory of Statistical Inference and Informatio.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.