An empirical Bayes derivation of best linear unbiased predictors

Let (Y1,θ1),…,(Yn,θn) be independent real-valued random vectors with Yi, given θi, is distributed according to a distribution depending only on θi for i=1,…,n. In this paper, best linear unbiased predictors (BLUPs) of the θi's are investigated. We show that BLUPs of θi's do not exist in certain situations. Furthermore, we present a general empirical Bayes technique for deriving BLUPs

Bayes and empirical Bayes estimation with errors in variables

Suppose that the random variable X is distributed according to exponential families of distributions, conditional on the parameter [theta]. Assume that the parameter [theta] has a (prior) distribution G. Because of the measurement error, we can only observe Y = X + [var epsilon], where the measurement error [theta] is independent of X and has a known distribution. This paper considers the squared error loss estimation problem of [theta] based on the contaminated observation Y. We obtain an expression for the Bayes estimator when the prior G is known. For the case G is completely unknown, an empirical Bayes estimator is proposed based on a sequence of observations Y1, Y2,...,Yn, where Yi's are i.i.d. according to the marginal distribution of Y. It is shown that the proposed empirical Bayes estimator is asymptotically optimal.Bayes Empirical Bayes Squared error loss estimation Kernel density estimates Asymptotically optimal

Asymptotically pointwise optimal allocation rules in Bayes sequential estimation

In this paper, the concept of asymptotic pointwise optimality in a single sequence of random variables provided by Bickel and Yahav [Bickel, P.J., Yahav, J.A., 1967. Asymptotically pointwise optimal procedures in sequential analysis. In: Proc Fifth Berkeley Symp. Math Statist. Prob. 1. University of California Press, pp. 401-413] is extended to more than one sequence of random variables. The Bayesian sequential estimation problem for one-parameter exponential families is considered and an asymptotically pointwise optimal rule, which includes a sequential allocation procedure and a stopping time, is provided. Some properties of asymptotic optimality are also obtained for the rules with and without using the prior information.