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November, 1976 Asymptotic Solutions to the Two State Component Compound Decision Problem, Bayes Versus Diffuse Priors on Proportions
Dennis C. Gilliland, James Hannan, J. S. Huang
Ann. Statist. 4(6): 1101-1112 (November, 1976). DOI: 10.1214/aos/1176343645

Abstract

Gilliland and Hannan (1974, Section 3) consider a general finite state compact risk component and reduce the problem of treating the asymptotic excess compound risk of Bayes compound rules to the question of $L_1$ consistency of certain induced estimators. This present paper considers the two state case and for several classes of diffuse symmetric priors on proportions establishes the $L_1$ consistency with rate. The rate $O(n^{-\frac{1}{2}})$ uniform in state sequences is shown for the uniform prior giving strong affirmation to the asymptotic form of a conjecture by Robbins (1951). The same or logarithmically weakened rate is shown for symmetric priors which are $\Lambda$-mixtures for several classes of $\Lambda$. A corollary shows a nonnull consistency, without regularity conditions, of a maximum likelihood estimator.

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Dennis C. Gilliland. James Hannan. J. S. Huang. "Asymptotic Solutions to the Two State Component Compound Decision Problem, Bayes Versus Diffuse Priors on Proportions." Ann. Statist. 4 (6) 1101 - 1112, November, 1976. https://doi.org/10.1214/aos/1176343645

Information

Published: November, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0344.62007
MathSciNet: MR448647
Digital Object Identifier: 10.1214/aos/1176343645

Subjects:
Primary: 62C25
Secondary: 62F10

Keywords: Bayes compound procedures , consistency of maximum likelihood estimator , consistency of posterior mean , consistent estimator of proportion , Two state compound decision problem

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 6 • November, 1976
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