Multivariate Contaminated Normal Censored Regression Model: Properties and Maximum Likelihood Inference

The Multivariate Contaminated Normal (MCN) distribution which contains two extra parameters with respect to parameters of the multivariate normal distribution, one for controlling the proportion of mild outliers and the other for specifying the degree of contamination, has been widely applied in robust statistics in the case of elliptically heavy-tailed empirical distributions. This article extends the MCN model to data with possibly censored values due to limits of quantification, referred to as the MCN with censoring (MCN-C) model, and further establishes the censored multivariate linear regression model where the random errors have the MCN distribution, named as the MCN censored regression (MCN-CR) model. Two computationally feasible Expectation Conditional Maximization (ECM) algorithms are developed for maximum likelihood estimation of MCN-C and MCN-CR models. An information-based method is used to approximate the standard errors of location parameters and regression coefficients. The capability and effectiveness of the MCN-C and MCN-CR models are illustrated via two real-data examples. A simulation study is conducted to investigate the superiority of the proposed models in terms of fit, accuracy of parameter estimation and censored data recovery as compared with classical approaches. Supplementary materials for this article are available online.

多元污染正态（Multivariate Contaminated Normal，MCN）分布相较于多元正态分布多两个额外参数，其一用于控制轻度异常值的占比，其二用于设定污染程度，在椭圆重尾经验分布场景下的稳健统计研究中已得到广泛应用。本文将MCN模型推广至因定量限导致存在可能删失值的数据集，将其称为带删失的MCN模型（MCN-C）；进一步构建了随机误差服从MCN分布的删失多元线性回归模型，命名为MCN删失回归（MCN-CR）模型。针对MCN-C与MCN-CR模型的极大似然估计，本文提出了两种计算可行的期望条件最大化（Expectation Conditional Maximization，ECM）算法。本文采用基于信息的方法近似估计位置参数与回归系数的标准误。通过两个真实数据集示例，验证了MCN-C与MCN-CR模型的性能与有效性。本文开展了模拟仿真研究，对比经典方法，从拟合效果、参数估计精度与删失数据恢复能力三个维度，验证了所提模型的优越性。本文的补充材料可在线获取。