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December, 1984 All Admissible Linear Estimators of the Mean of a Gaussian Distribution on a Hilbert Space
Avi Mandelbaum
Ann. Statist. 12(4): 1448-1466 (December, 1984). DOI: 10.1214/aos/1176346803

Abstract

We consider linear estimators for the mean $\theta$ of a Gaussian distribution $N(\theta, C)$ on a Hilbert space, when the covariance operator $C$ is known. It was argued in a previous work that the natural class of linear estimators is the class of measurable linear transformations. Using the simplest quadratic loss we prove that the linear estimator $L$ is admissible if and only if the operator $C^{-1/2}LC^{1/2}$ is Hilbert-Schmidt, self-adjoint, its eigenvalues are all between 0 and 1 and two are equal to 1 at the most. As an application of the general theory, we investigate some linear estimators for the drift function of a Brownian motion.

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Avi Mandelbaum. "All Admissible Linear Estimators of the Mean of a Gaussian Distribution on a Hilbert Space." Ann. Statist. 12 (4) 1448 - 1466, December, 1984. https://doi.org/10.1214/aos/1176346803

Information

Published: December, 1984
First available in Project Euclid: 12 April 2007

zbMATH: 0558.62009
MathSciNet: MR760699
Digital Object Identifier: 10.1214/aos/1176346803

Subjects:
Primary: 62C15
Secondary: 60G15, 62C07, 62H12, 62H99

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 4 • December, 1984
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