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.