Joint Robust Variable Selection of Mean and Covariance Model via Shrinkage Methods

A valuable and robust extension of the traditional joint mean and the covariance models when data subject to outliers and/or heavy‐tailed outcomes can be achieved using the joint modelling of location and scatter matrix of the multivariate t‐distribution. This model encompasses three models in itself, and the number of unknown parameters in the covariance model increases quadratically with the matrix size. As a result, selecting the important variables becomes a crucial aspect to consider. In this context, the variable selection combined with the parameter estimation is considered under the normality assumption. However, because of the non‐robustness of the normal distribution, the resulting estimators will be sensitive to outliers and/or heavy taildness in the data. This paper has two objectives to overcome these problems. The first is to obtain the maximum likelihood estimates of the parameters and propose an expectation‐maximisation type algorithm as an alternative to the Fisher scoring algorithm in the literature. We also consider simultaneous parameter estimation and variable selection in the multivariate t‐joint location and scatter matrix models. The consistency and oracle properties of the regularised estimators are also established. Simulation studies and real data analysis are provided to assess the performance of the proposed methods.