A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications

TL;DR

This paper develops a symmetric matrix-variate normal approximation for the Wishart distribution, providing explicit asymptotic expansions, bounds on total variation, and a new density estimator with proven properties, enhancing statistical inference on positive definite matrices.

Contribution

It introduces the first explicit local normal approximation for the Wishart distribution, along with asymptotic expansions, bounds, and a novel density estimator with theoretical guarantees.

Findings

Derived a precise asymptotic expansion for the Wishart-to-normal density ratio.

Established an upper bound on total variation distance between Wishart and normal measures.

Proposed a new density estimator with proven bias, variance, and asymptotic normality.

Abstract

The noncentral Wishart distribution has become more mainstream in statistics as the prevalence of applications involving sample covariances with underlying multivariate Gaussian populations as dramatically increased since the advent of computers. Multiple sources in the literature deal with local approximations of the noncentral Wishart distribution with respect to its central counterpart. However, no source has yet developed explicit local approximations for the (central) Wishart distribution in terms of a normal analogue, which is important since Gaussian distributions are at the heart of the asymptotic theory for many statistical methods. In this paper, we prove a precise asymptotic expansion for the ratio of the Wishart density to the symmetric matrix-variate normal density with the same mean and covariances. The result is then used to derive an upper bound on the total variation between the corresponding probability measures and to find the pointwise variance of a new density estimator on the space of positive definite matrices with a Wishart asymmetric kernel. For the sake of completeness, we also find expressions for the pointwise bias of our new estimator, the pointwise variance as we move towards the boundary of its support, the mean squared error, the mean integrated squared error away from the boundary, and we prove its asymptotic normality.