Convex models, MLE and misspecification
Convex models, MLE and misspecification
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DOI：
10.1214/aos/996986503
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发表时间：
2001-02
影响因子：
4.5
通讯作者：
V. Patilea
中科院分区：
文献类型：
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作者：
V. Patilea
我们分析当独立数据生成独立数据的真实分布不一定属于模型时，在凸模的模型中分析了最大似然估计器（MLE）的渐近行为。受Hellinger距离及其特性的启发，我们引入了一个差异（对比功能），该家族允许对良好和刻不清的凸模型进行统一的处理。 MLE相对于我们的分歧的收敛性和收敛速率是从这些差异所满足的不平等中获得的，以及经验过程理论的结果（大数量和最大不平等的统一定律）。作为一种特殊情况，我们恢复了在明确指定的凸模型中MLE的现有结果。考虑了四个示例：离散分布的混合物，单调密度，下降率分布和有限维参数模型。
We analyze the asymptotic behavior of maximum likelihood estimators (MLE) in convex dominated models when the true distribution generating the independent data does not necessarily belong to the model. Inspired by the Hellinger distance and its properties, we introduce a family of divergences (contrast functions) which allow a unified treatment of well- and misspecified convex models. Convergence and rates of convergence of the MLE with respect to our divergences are obtained from inequalities satisfied by these divergences and results from empirical process theory (uniform laws of large numbers and maximal inequalities). As a particular case we recover existing results for Hellinger convergence of MLE in well-specified convex models. Four examples are considered: mixtures of discrete distributions, monotone densities, decreasing failure rate distributions and a finite-dimensional parametric model.