Estimation of the entropy of a multivariate normal distribution

Abstract

Motivated by problems in molecular biosciences wherein the evaluation of entropy of a molecular system is important for understanding its thermodynamic properties, we consider the efficient estimation of entropy of a multivariate normal distribution having unknown mean vector and covariance matrix. Based on a random sample, we discuss the problem of estimating the entropy under the quadratic loss function. The best affine equivariant estimator is obtained and, interestingly, it also turns out to be an unbiased estimator and a generalized Bayes estimator. It is established that the best affine equivariant estimator is admissible in the class of estimators that depend on the determinant of the sample covariance matrix alone. The risk improvements of the best affine equivariant estimator over the maximum likelihood estimator (an estimator commonly used in molecular sciences) are obtained numerically and are found to be substantial in higher dimensions, which is commonly the case for atomic coordinates in macromolecules such as proteins. We further establish that even the best affine equivariant estimator is inadmissible and obtain Stein-type and Brewster–Zidek-type estimators dominating it. The Brewster–Zidek-type estimator is shown to be generalized Bayes.