We investigate existence and properties of discrete mixture representations $P_\theta =\sum_{i\in E} w_\theta(i) \, Q_i$ for a given family $P_\theta$, $\theta\in\Theta$, of probability measures. The noncentral chi-squared distributions provide a classical example. We obtain existence results and results about geometric and statistical aspects of the problem, the latter including loss of Fisher information, Rao-Blackwellization, asymptotic efficiency and nonparametric maximum likelihood estimation of the mixing probabilities.
翻译:我们调查了离散混合物在概率测量上的存在和特性,表现为 $P ⁇ theta ⁇ sumíi\ in E} w ⁇ theta (i) \, i $i$, 对于特定家庭来说, $P ⁇ theta$, $\theta\ in\ theta$, $\ theta$, uncentral chiqured 分布提供了典型的例子。我们获得了关于该问题的几何和统计方面的结果和结果,后者包括渔渔渔资料的丢失、Rao-Blackwelliz化、无药效和混合概率的非参数最大可能性估计。