Wishart矩阵二次型初步理论及其应用

THE ESSENTIAL THEORY OF THE QUADRATIC FORMS OF THE WISHART MATRIX AND ITS APPLICATION

设A～W_m（n,Σ）,D为m×m对称阵,称B=A＇DA为Wishart矩阵二次型,本文讨论了Wishart矩阵二次型的分布密度及其各种初步性质,并将其推广到椭球等高分布族,最后用二次型理论解决了求正态总体协差阵Σ的某一类Bayes估计的问题。

Suppose that A is Wm(n,), D is mxm symmetric matrix, B = A'DA is called the quadratic forms of the 'Wishart matrix. In this paper, the density function, the expectation, and the other essential properties of B are obtained. The above results can also be extended into spherical and elliptical distributions. At last, the theory of the quatlratic forms of Wishart matrix is applied to the Bayes estimation of the coyariance matrix 2 in normal population.