In this work, we develop Stein's method for the Wishart distribution on the cone of positive definite matrices. We establish the basic ingredients of a Wishart Stein framework: we derive an extended-generator-based Stein characterization from the Wishart diffusion process, identify the corresponding transition semigroup through the noncentral Wishart law, provide an explicit semigroup representation for the solution of the Stein equation, and obtain regularity estimates for the solution. The new methodology is demonstrated in four applications: (i) an order $n^{-1}$ bound, for smooth test functions, for the Wishart approximation of uncentered group-mean scatter matrices in MANOVA; (ii) a quantitative multivariate Satterthwaite approximation; (iii) local/integrated De Bruijn identities and logarithmic Sobolev inequalities for the Wishart measure; and (iv) Stein's method of moments for the shape and scale parameters, including structured scale estimation.

翻译：本文在正定矩阵锥上发展了Wishart分布的Stein方法。我们建立了Wishart Stein框架的基本要素：从Wishart扩散过程推导了基于扩展生成元的Stein刻画，通过非中心Wishart律识别了相应的转移半群，给出了Stein方程解的显式半群表示，并获得了解的正则性估计。新方法通过四个应用得到验证：（i）对于MANOVA中未中心化组均值散度矩阵的Wishart逼近，针对光滑检验函数给出$n^{-1}$阶界限；（ii）定量多元Satterthwaite逼近；（iii）Wishart测度的局部/积分De Bruijn恒等式与对数Sobolev不等式；以及（iv）形状和尺度参数的Stein矩方法，包含结构化尺度估计。