Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators
Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators
复制标题
最大似然估计量收敛到正态性速率的最优阶均匀和非均匀界限
DOI：
10.1214/17-ejs1264
复制
发表时间：
2017
影响因子：
1.1
通讯作者：
I. Pinelis
中科院分区：
文献类型：
--
作者：
I. Pinelis
众所周知，在一般正则性条件下，极大似然估计（MLE）的分布是渐近正态的。最近，在所谓的有界Wasserstein距离中，获得了MLE分布接近正态分布的最佳阶数O（1 /n）的界[2，1]，其中n是样本大小。然而，相应的Kolmogorov距离的界仅为O（1 /n1/ 4）阶.本文给出了极大似然估计的分布在Kolmogorov距离上接近正态分布的最优阶数O（1 /n）的界，以及它们的非一致界，这些界在极大似然估计分布的尾区更有效。这些结果部分基于之前获得的多元delta方法收敛到正态性速度的一般最优阶界限。关键的观察是，在自然条件下，极大似然估计可以被两个足够光滑的独立随机向量之和的函数足够紧密地包围，这使得delta方法是适用的。在现有的文献中，似乎一般情况下MLE的非均匀边界没有先例; Pinelis和Molzon最近处理了一个特例[20]。所得结果可推广到M -估计.
: It is well known that, under general regularity conditions, the distribution of the maximum likelihood estimator (MLE) is asymptotically normal. Very recently, bounds of the optimal order O (1 / √ n ) on the close- ness of the distribution of the MLE to normality in the so-called bounded Wasserstein distance were obtained [2, 1], where n is the sample size. How- ever, the corresponding bounds on the Kolmogorov distance were only of the order O (1 /n 1 / 4 ). In this paper, bounds of the optimal order O (1 / √ n ) on the closeness of the distribution of the MLE to normality in the Kol- mogorov distance are given, as well as their nonuniform counterparts, which work better in tail zones of the distribution of the MLE. These results are based in part on previously obtained general optimal-order bounds on the rate of convergence to normality in the multivariate delta method. The cru- cial observation is that, under natural conditions, the MLE can be tightly enough bracketed between two smooth enough functions of the sum of in- dependent random vectors, which makes the delta method applicable. It appears that the nonuniform bounds for MLEs in general have no prece- dents in the existing literature; a special case was recently treated by Pinelis and Molzon [20]. The results can be extended to M -estimators.