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.

翻译：在统计中,非中Wishart分布已变得更加主流,因为自计算机出现以来,与基础多变Gaussia人口有关的抽样共变应用的普及程度随着计算机的出现而急剧增加。文献中的多种来源涉及非中Wishart分布相对于其中央对应方的局部近似值。然而,没有任何来源为(中央)Wishart分布以正常的类似值为基础,为(中央)Wishart分布制定明确的本地近似值,这一点很重要,因为Gaussian分布是许多统计方法的无药可治理论的核心。在本文中,我们证明Wishart密度与对称矩阵变异正常密度之比的比例是精确的无药可治的扩大。结果被用来对相应的概率计量尺度之间的全面差异形成一个上限值,并找出与Wishart不对称内核的正确定矩阵空间的新密度估计值之间的点差值差异。为了完整起见,我们还发现我们新的估测点偏差的表达方式,即Wishart不对称密度密度密度与对称矩阵中基质矩阵的平差,即我们向正平方边界的平方边界的正确,我们向平方边界的平方表示。