A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications
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.
Additional Information© 2021 Elsevier Inc. Received 19 June 2021, Revised 17 November 2021, Accepted 17 November 2021, Available online 8 December 2021. First, I would like to thank Donald Richards for his indications on how to calculate the moments in Lemma 1. I also thank the Editor, the Associate Editor and the referees for their insightful remarks which led to improvements in the presentation of this paper. The author is supported by postdoctoral fellowships from the NSERC (PDF) and the FRQNT (B3X supplement and B3XR). CRediT authorship contribution statement: Frédéric Ouimet: Writing of the original draft and editing, Review of the literature, Conceptualization, Theoretical results and proofs, Responsible for all the sections.
Submitted - 2104.04882.pdf
Supplemental Material - 1-s2.0-S0047259X21001913-mmc1.r