A bounded influence regression estimator based on the statistics of the hat matrix
A bounded influence regression estimator based on the statistics of the hat matrix
复制标题
基于帽子矩阵统计的有界影响回归估计器
DOI：
10.1111/1467-9876.00406
复制
发表时间：
2003
期刊：
影响因子：
--
通讯作者：
D. Thomson
中科院分区：
文献类型：
--
作者：
A. Chave;D. Thomson
摘要许多地球物理回归问题需要分析大型（超过104个值）数据集，并且由于数据可能代表具有广泛变化的统计特性的并发自然过程的混合物，响应和预测变量的污染是常见的。现有的有界影响或高崩溃点估计经常缺乏消除极其有影响的数据和/或处理大数据集的计算效率的能力。提出了一种新的有界影响估计器，它结合了正常数据的高渐近效率，污染数据的高崩溃点行为和大数据集的计算简单性。该算法结合了一个标准的M-估计器，以降低与极端回归残差相对应的数据的权重，并根据帽子矩阵对角元素的统计数据去除过度影响的预测值（杠杆点）。为此，复多变量高斯预测器数据的帽子矩阵对角元素pii的精确分布被示为β（pii，m，N-m），其中N是数据的数量，m是参数的数量。 极光区大地电磁研究的真实的地球物理数据表现出严重的离群值和杠杆点污染被用来说明估计的性能。这些例子还展示了通过分位数-分位数图来诊断稳健回归问题的残差和帽子矩阵分布的实用性。
Summary. Many geophysical regression problems require the analysis of large (more than 104 values) data sets, and, because the data may represent mixtures of concurrent natural processes with widely varying statistical properties, contamination of both response and predictor variables is common. Existing bounded influence or high breakdown point estimators frequently lack the ability to eliminate extremely influential data and/or the computational efficiency to handle large data sets. A new bounded influence estimator is proposed that combines high asymptotic efficiency for normal data, high breakdown point behaviour with contaminated data and computational simplicity for large data sets. The algorithm combines a standard M‐estimator to downweight data corresponding to extreme regression residuals and removal of overly influential predictor values (leverage points) on the basis of the statistics of the hat matrix diagonal elements. For this, the exact distribution of the hat matrix diagonal elements pii for complex multivariate Gaussian predictor data is shown to be β(pii, m, N−m), where N is the number of data and m is the number of parameters. Real geophysical data from an auroral zone magnetotelluric study which exhibit severe outlier and leverage point contamination are used to illustrate the estimator's performance. The examples also demonstrate the utility of looking at both the residual and the hat matrix distributions through quantile–quantile plots to diagnose robust regression problems.