Empirical Bayes Estimation of a Location Parameter

Abstract

Assume that in independent two-dimensional random vectors (Λι, θ\) ,..., (ΧΝ,ΘΝ), each Θϊ is distributed according to some unknown prior distribution density g, and that , given θ,, Xi has the conditional density function q(x -Θ,), i = 1,..., TV. In each pair the first component is observable, but the second is not. After the (JV+l)-th pair (X^r+i, is obtained, the objective is to construct the empirical Bayes estimator of a polynomial f(0w + 1) = with given coefficients bj. In the paper we derive the empirical Bayes estimator of b(9) without any parametric assumptions on g. The upper bound for the mean squared error is obtained. The lower bound for the mean squared error over the class of all possible empirical Bayes estimators is also derived. It is shown that the estimators constructed in the paper have the optimal or nearly optimal convergence rates. Examples for familiar families of conditional distributions are considered.

On empirical Bayes estimation of multivariate regressioncoefficient

“…In the present problem, the regret of φ is equal to R n (y) f (y)dy with R n (y) given by (4.7). However, Pensky [11] pointed out at least two advantages of using (4.7) compared to the regret. First, R n (y) enables one to calculate the mean squared error for the given observation Y n+1 = y which is the interesting quantity.…”

“…In this paper, we study the performance of our EB estimator using the "mean squared error" (MSE) criterion. Our rationale to study the MSE of our EB estimator is motivated by the work of Pensky [11], where she has argued that MSE is a more appropriate criterion as far as applications are concerned. We will construct an EB estimator of β (for the unknown σ 2 case) based on some improved estimators of multivariate normal mixture density and its first partial derivatives.…”

Empirical Bayes Testing for a Nonexponential Family Distribution

“…Pensky (1997) studied empirical Bayes estimation of the location-parameter in a location-parameter family distribution. Li and Gupta (2001) studied an empirical Bayes test in a truncation parameter family distribution and showed that the proposed empirical Bayes test achieves the optimal rate of convergence.…”

On the lower bounds for mean square error of empirical bayes estimators

“…Various methods of empirical Bayes estimation have been considered and estimates with different convergence rates have been obtained (see, for instance, [4,6,7,13,[15][16][17][18]). In this connection the question becomes significant: what is the best convergence rate that can be obtained for a [12]) and develop the ideas of her papers [9][10][11]. In this connection the question becomes significant: what is the best convergence rate that can be obtained for a [12]) and develop the ideas of her papers [9][10][11].…”