  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.
