An Empirical Bayes Estimation Problem with Nonidentical Components Involving Normal Distributions
Titel:
An Empirical Bayes Estimation Problem with Nonidentical Components Involving Normal Distributions
Auteur:
O'Bryan, T. Susarla, V.
Verschenen in:
Communications in statistics
Paginering:
Jaargang 4 (1975) nr. 11 pagina's 1033-1042
Jaar:
1975
Inhoud:
Let {an}, {bn}, and [image omitted] be known sequences of real numbers with an ≠ 0 and [image omitted] 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 [image omitted] . 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 zero. Then, with Rn+1(G) denoting the infimum 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.
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