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

Author

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  • Berger, James
  • Wolpert, Robert

Abstract

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.

Suggested Citation

  • Berger, James & Wolpert, Robert, 1983. "Estimating the mean function of a Gaussian process and the Stein effect," Journal of Multivariate Analysis, Elsevier, vol. 13(3), pages 401-424, September.
    • Handle: RePEc:eee:jmvana:v:13:y:1983:i:3:p:401-424
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