混合分布的识别性及其应用

IDENTIFIABILITY OF MULTIVARIATE MIXED DISTRIBUTI(?) AND ITS APPLICATIONS

设(Y_1,Y_2)是非负的二维随机变量,有联合分布函数F(y_1,y_2),设Z=min(Y_1,Y_2),定义随机变量I=1;2;3,分别对应于Y_1<Y_2;Y_1>Y_2;Y_1=Y_2时.记p_i=P(I=i),f_i(·)为给定I=i时Z的条件密度(i=1;2;3).给定可识最小值(Z,I)的联合密度函数P_iJ_i(·)(i=1;2;3),得到F(·,·)是混合分布时,F(·,·)用P_if_i(·)来表示的一个显式表达式.对三维及以上情形,得到类似的结果.特别,把识别基本定理应用于多元Marshall-Olkin型指数分布的参数估计,得到了基于观察值(Z_jI_j)(j=1,2,…,n)的参数的极大似然估计及矩估计,并且证明了极大似然估计具有联合完备充分的,渐近无偏的,均方误差一致的,及渐近正态性特性,修正估计恰好是Arnold估计,它是唯一的一致最小方差无偏估计,且同时是渐近有效估计,本文还指出了矩估计方法的不唯一性.

Let Y_1,…,Y_m be m random variables with joint distribution function F(y_1,y_2,…,y_m) (y_1 > 0, …,y_m > 0). LetZ = min (Y_1, …,Y_m), and define I= {j_1,…,j_s} if Y_(j_1)=Y_(j_2) =…= Y_(j_s) < Y_j (for all j ≠ i_u, u = 1,…,s). The problem of identifying the distribution function F(y_1,y_2,…, y_m), with the given joint density of (Z,I), is considered when F is a mixed distribution. Without any further regularity assumptions, the distribution F is expressed explicitly ar. functionals of p{j_1,…,j_s}f{j_1,…,j_s}(·)(1≤j_1<j_2 <…<j_sm,s=1,…,m). As applications, suppose (Y_(1j),…, Y_(my)) (j=1,…,m) is a sample from the Multiveriate Exponential Distribution of the MarshallOlkin form with the parameters λ_(j_1…j_s)(1≤j_1<…<j_s≤m,s= 1,…,m), the likihood function of the observations (Z_j,I_j) (j = 1,…,n) is given. It's shown that the maxium likelihood estimator of the λ_(j_1…j_s) s is asymptotically unbiased, joint complete and sufficient, mean squared error consistent and has the asymptotic normality property. The adjusted estimator is just the Arnold's estimator. It is pointed out that the mothed of moments estimators is not unique.