## The Annals of Statistics

- Ann. Statist.
- Volume 12, Number 4 (1984), 1448-1466.

### All Admissible Linear Estimators of the Mean of a Gaussian Distribution on a Hilbert Space

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 12, Number 4 (1984), 1448-1466.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346803

**Digital Object Identifier**

doi:10.1214/aos/1176346803

**Mathematical Reviews number (MathSciNet)**

MR760699

**Zentralblatt MATH identifier**

0558.62009

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62C15: Admissibility

Secondary: 62C07: Complete class results 62H12: Estimation 60G15: Gaussian processes 62H99: None of the above, but in this section

**Keywords**

Linear estimators admissibility Gaussian measures on a Hilbert space measurable linear transformations Brownian motion Orenstein-Uhlenbeck process

#### Citation

Mandelbaum, Avi. All Admissible Linear Estimators of the Mean of a Gaussian Distribution on a Hilbert Space. Ann. Statist. 12 (1984), no. 4, 1448--1466. doi:10.1214/aos/1176346803. https://projecteuclid.org/euclid.aos/1176346803