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.
Publié le : 1976-11-14
Classification:
Two state compound decision problem,
Bayes compound procedures,
consistent estimator of proportion,
consistency of maximum likelihood estimator,
consistency of posterior mean,
62C25,
62F10
@article{1176343645,
author = {Gilliland, Dennis C. and Hannan, James and Huang, J. S.},
title = {Asymptotic Solutions to the Two State Component Compound Decision Problem, Bayes Versus Diffuse Priors on Proportions},
journal = {Ann. Statist.},
volume = {4},
number = {1},
year = {1976},
pages = { 1101-1112},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343645}
}
Gilliland, Dennis C.; Hannan, James; Huang, J. S. Asymptotic Solutions to the Two State Component Compound Decision Problem, Bayes Versus Diffuse Priors on Proportions. Ann. Statist., Tome 4 (1976) no. 1, pp. 1101-1112. http://gdmltest.u-ga.fr/item/1176343645/