Premium
EMPIRICAL BAYES ESTIMATION WITH NON‐IDENTICAL COMPONENTS. CONTINUOUS CASE.
Author(s) -
O'Bryan Thomas E.,
Susarla V.
Publication year - 1977
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1977.tb01278.x
Subject(s) - bayes' theorem , mathematics , estimator , sequence (biology) , exponential family , asymptotically optimal algorithm , combinatorics , exponential function , bayes' rule , empirical distribution function , bayes estimator , exponential distribution , component (thermodynamics) , distribution (mathematics) , discrete mathematics , statistics , bayes factor , algorithm , mathematical analysis , bayesian probability , genetics , physics , biology , thermodynamics
In this paper a variant of the standard empirical Bayes estimation problem is considered where the component problems in the sequence are not identical in that the conditional distribution of the observations may vary with the component problems. Let {(Θ n , X n )} be a sequence of independent random vectors where Θ n ˜ G and, given Θ n =Θ n , X n ‐P Θ,m(n) with {m(n)} a sequence of positive integers where m(n)≤m̄ < ∞ for all n. With P Θ,m in a continuous exponential family of distributions, asymptotically optimal empirical Bayes procedures are exhibited which depend on uniform approximations of certain functions on compact sets by polynomials in e Θ . Such approximations have been explicitly calculated in the case of normal and gamma families. In the case of normal families, asymptotically optimal linear empirical Bayes estimators in the class of all linear estimators are derived and shown to have rate O(n ‐1/2 ).