An empirical bayes estimation problem with nonidentical components involving normal distributions
- 1 January 1975
- journal article
- research article
- Published by Taylor & Francis in Communications in Statistics
- Vol. 4 (11) , 1033-1042
- https://doi.org/10.1080/03610927508827311
Abstract
Let {an}, {bn}, and be known sequences of real numbers with an ≠ 0 and and let {(θn, Xn)} be a sequence of independent random vectors where the θn are iid G and unobservable and, given θn = θ, Xn has the univariate normal density with mean an θ + bn and variance The first part of the paper exhibits estimators for the density of Xn and its kth derivative, k = 1,2,…, and obtains rates at which the mean-squared errors go to tsro Then, with Rn+1(G) denoting the infinum Bayes risk for the squared error loss estimation of θn+1 using Xn+1, the second part of the paper exhibits asymptotically optimal empirical Bayes rules tn (X1,…, Xn; Xn+1) and obtains rates at which E(tn − θn+1)2 −Rn+1(G)−0.Keywords
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