齐性球对称分布的特征

ON A CHARACTERIZATION OF HOMOGENEOUS SPHERICAL DISTRIBUTION

设随机变量ｘ＝（ｘ１，…，ｘｎ）′服从球对称分布，密度函数是ｃｎ［ｆ（ｘ′ｘ）］ｇ（ｎ）．若分量ｘ（１）＝（ｘ１，…，ｘｐ）′的边缘密度函数是ｃｐ［ｆ（ｘ（１）′ｘ（１））］ｇ（ｐ），１≤ｐ≤ｎ－１，ｃｐ和ｇ（ｐ）是只依赖于ｐ的常数，则称ｘ具有齐性球对称分布．本文讨论了齐性球对称分布的特征，确定了函数ｆ（）、ｇ（）的形式．最后。

Let random vector x=(x 1,…,x n)′ has a spherical distribution with joint density function c n g(n) , where c n and g(n) are constants. The random vector x is said to have a homogeneous spherical distribution if for all 1≤p≤n-1 , the marginal density function of x (1) =(x 1,…,x p)′ is c p[f(x (1) ′x (1) ] g(p) , where c p and g(p) are constants. In this paper, we discuss the characterization of homogeneous spherical distribution, obtain the forms of f(·) and g(·) . Finally, we extend some results to the case of multivariate spherical distribution.