Maximum likelihood estimators based on the block maxima method
Maximum likelihood estimators based on the block maxima method
复制标题
基于块极大值方法的最大似然估计器
DOI：
10.3150/18-bej1032
复制
发表时间：
2017
期刊：
影响因子：
1.5
通讯作者：
Ana Ferreira
中科院分区：
文献类型：
--
作者：
C. Dombry;Ana Ferreira
极值指数是单变量极值理论中的一个基本参数。它捕获分布的尾部行为，并且在观测数据之外的外推中处于中心位置。在其他半参数方法（如流行的Hill估计）中，块极大值（BM）和峰值超过阈值（POT）方法被广泛用于评估极值指数和相关的归一化常数。给出了基于BM方法的极大似然估计的渐近理论。我们的主要结果是具有非平凡偏差的MLE的渐近正态性取决于极值指标和所谓的二阶参数。我们的方法结合了似然过程的渐近展开和块最大值的经验分位数过程。结果允许通过其渐近方差，偏差和最优均方误差完成EVT中最常见的半参数估计（基于POT或BM方法的MLE和概率加权矩估计）的比较。
The extreme value index is a fundamental parameter in univariate Extreme Value Theory (EVT). It captures the tail behavior of a distribution and is central in the extrapolation beyond observed data. Among other semi-parametric methods (such as the popular Hill's estimator), the Block Maxima (BM) and Peaks-Over-Threshold (POT) methods are widely used for assessing the extreme value index and related normalizing constants. We provide asymptotic theory for the maximum likelihood estimators (MLE) based on the BM method. Our main result is the asymptotic normality of the MLE with a non-trivial bias depending on the extreme value index and on the so-called second order parameter. Our approach combines asymptotic expansions of the likelihood process and of the empirical quantile process of block maxima. The results permit to complete the comparison of most common semi-parametric estimators in EVT (MLE and probability weighted moment estimators based on the POT or BM methods) through their asymptotic variances, biases and optimal mean square errors.