反Dirichlet分布,Fang-xu分布,Dirichlet分布的特征

TWO CHARACTERIZATIONS OF THE DIRICHLET DISTRIBUTIONS,THE INVERTED DIRICHLET DISTRIBUTIONS AND THE FANG—XU DISTRIBUTIONS

设 X=(X_1,X_2,…,X_n),Y=(Y_1,Y_2,…,Y_n),Z=(Z_1,Z_2,…,Z_n)分别是随机向量,X_i>0,-∞<Y_i<+∞,0<Z_j<1,sum from i=1 to n Z_i<1(i=1,2,…,n),本文分别给出并证明了 X 服从反 Dirichlet 分布或它的极限,Y 服从 Fang—Xu 分布或它的极限,Z服从 Dirichlet 分布或它的极限的两个等价的充分必要条件。并由此给出了关于齐次条件的一个统计证明。

Let n≥2,X=(X_1,…,X_n),Y=(Y_1,…,Y_n),Z=(Z_1,…,Z_n)be random vectors with X_i>0,—∞<Y_i<+∞,0<Z_i<1,i=1,…,N,sum from i=1 to n Z_<1,without any further regu- larity assumptions,each of two conditions is shown to be necessary and sufficient for X,Y,Z to be distributed according to the Inverted Dirichlet distribution,the Fang—Xu distribution,the Dirichlet distribution,or the limit of their distributions respectively.As an application,a statistical justification for the homogeneity requirements is provided