Estimating the mean function of a Gaussian process and the Stein effect

The problem of global estimation of the mean function [theta](·) of a quite arbitrary Gaussian process is considered. The loss function in estimating [theta] by a function a(·) is assumed to be of the form L([theta], a) = [integral operator] [[theta](t) - a(t)]2[mu](dt), and estimators are evaluated in terms of their risk function (expected loss). The usual minimax estimator of [theta] is shown to be inadmissible via the Stein phenomenon; in estimating the function [theta] we are trying to simultaneously estimate a larger number of normal means. Estimators improving upon the usual minimax estimator are constructed, including an estimator which allows the incorporation of prior information about [theta]. The analysis is carried out by using a version of the Karhunen-Loéve expansion to represent the original problem as the problem of estimating a countably infinite sequence of means from independent normal distributions.