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Uniformly best invariant stopping rules

Published online by Cambridge University Press:  14 July 2016

Steinar Engen  and
Eva Seim
Steinar Engen
Affiliation:
University of Trondheim
Eva Seim*
Affiliation:
University of Trondheim
*
∗Postal address: Department of Mathematics and Statistics, University of Trondheim, Den allmennvitenskapelige høgskolen, Trondheim, N-7055 Dragvoll, Norway.
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Abstract

The class of stopping rules for a sequence of i.i.d. random variables with partially known distribution is restricted by requiring invariance with respect to certain transformations. Invariant stopping rules have an intuitive appeal when the optimal stopping problem is invariant with respect to the actual gain function. Uniformly best invariant stopping rules are derived for the gamma distribution with known shape parameter and unknown scale parameter, for the uniform distribution with both endpoints unknown, and for the normal distribution with unknown mean and variance. Some comparisons with previously published results are made.

Keywords

OPTIMAL STOPPINGINVARIANCEGAMMA DISTRIBUTIONSUNIFORM DISTRIBUTIONNORMAL DISTRIBUTIONSECRETARY PROBLEM

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987 , pp. 77 - 87
Copyright
Copyright © Applied Probability Trust 1987 

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References

Bowerman, B. L. and Koehler, A. B. (1978) An optimal policy for sampling from uncertain distributions. Commun. Statist. A 7, 1041–1051.Google Scholar
Chow, Y. S., Robbins, H. and Siegmund, D. (1971) Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston.Google Scholar
Degroot, M. H. (1968) Some problems of optimal stopping. J. R. Statist. Soc. B 30, 108–122.Google Scholar
Freeman, P. R. (1983) The secretary problem and its extensions: a review. Internat. Statist. Rev. 51, 189–206.10.2307/1402748Google Scholar
Gilbert, J. P. and Mosteller, F. (1966) Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.Google Scholar
Johnson, N. L. and Kotz, S. (1970) Continuous Univariate Distributions – I, Distributions in Statistics. Wiley, New York.Google Scholar
Kiefer, J. (1957) Invariance, minimax sequential estimation, and continuous time processes. Ann. Math. Statist. 28, 573–601.Google Scholar
Lehmann, E. L. (1959) Testing Statistical Hypotheses. Wiley, New York.Google Scholar
Lindley, D. V. (1961) Dynamic programming and decision theory. Appl. Statist. 10, 39–52.10.2307/2985407Google Scholar
Sakaguchi, M. (1961) Dynamic programming of some sequential sampling design. J. Math. Anal. Appl. 2, 446–466.Google Scholar
Samuels, S. M. (1981) Minimax stopping rules when the underlying distribution is uniform. J. Amer. Statist. Assoc. 76, 188–197.Google Scholar
Shiryayev, A. N. (1978) Optimal Stopping Rules. Springer-Verlag, New York.Google Scholar
Stewart, T. J. (1978) Optimal selection from a random sequence with learning of the underlying distribution. J. Amer. Statist. Assoc. 73, 775–780.Google Scholar