球对称分布下的二次型
QUADRATIC FORMS UNDER SPHERICAL DISTRIBUTIONS
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摘要: 设$ \underset{N \times P}{X}=\binom{X_{(1)}}{X_{(2)}}$ 为球对称分布的矩阵,本文将证实如下命题的等价性:1.X(1)与X(2)相互独立;2.X′(1)X(1)与X′(2)X(2)相互独立;3.vecX依 $N(0, \nabla \otimes I)$ 分布,V是某个非负定阵。最后,在P（X=0）<1条件下,我们将关于二次型的Cochran定理推广至更一般的情形。Abstract: Let $ \underset{N \times P}{X}=\binom{X_{(1)}}{X_{(2)}}$ be a spherically Symmetric distributed matrix, we shall prove the equivalence of the following propositions: 1. X(1)andX(2)are mutually independent; 2. X'(1)X(1)andX'(2)X(2) are mutually independent; 3. Vec X is distributed as $N(0, \nabla \otimes I)$ for some non-negative matrix V. At last, under the ristriction P（X=0）＜1 we extend the classical Cochran’s theorem of quadratic formt to a more general fashion.
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