# Topics on Regularization of Parameters in Multivariate Linear Regression

## Abstract

My dissertation mainly focuses on the regularization of parameters in the multivariate linear regression under different assumptions on the distribution of the errors. It consists of two topics where we develop iterative procedures to construct sparse estimators for both the regression coefficient and scale matrices simultaneously, and a third topic where we develop a method for testing if the skewness parameter in the skew-normal distribution is parallel to one of the eigenvectors of the scale matrix.

In the first project, we propose a robust procedure for constructing a sparse estimator of a multivariate regression coefficient matrix that accounts for the correlations of the response variables. Robustness to outliers is achieved using heavy-tailed t distributions for the multivariate response, and shrinkage is introduced by adding to the negative log-likelihood l1 penalties on the entries of both the regression coefficient matrix and the precision matrix of the responses. Taking advantage of the hierarchical representation of a multivariate t distribution as the scale mixture of normal distributions and the EM algorithm, the optimization problem is solved iteratively where at each EM iteration suitably modified multivariate regression with covariance estimation (MRCE) algorithms proposed by Rothman, Levina and Zhu are used. We propose two new optimization algorithms for the penalized likelihood, called MRCEI and MRCEII, which differ from MRCE in the way that the tuning parameters for the two matrices are selected. Estimating the degrees of freedom when penalizing the entries of the matrices presents new computational challenges. A simulation study and real data analysis demonstrate that the MRCEII, which selects the tuning parameter of the precision matrix of the multiple responses using the Cp criterion, generally does the best among all methods considered in terms of the prediction error, and MRCEI outperforms the MRCE methods when the regression coefficient matrix is less sparse.

The second project is motivated by the existence of the skewness in the data for which the symmetric distribution assumption on the errors does not hold. We extend the procedure we have proposed to the case where the errors in the multivariate linear regression follow a multivariate skew-normal or skew-t distribution. Based on the convenient representation of skew-normal and skew-t as well as the EM algorithm, we develop an optimization algorithm, called MRST, to iteratively minimize the negative penalized log-likelihood. We also carry out a simulation study to assess the performance of the method and illustrate its application with one real data example.

In the third project, we discuss the asymptotic distributions of the eigenvalues and eigenvectors for the MLE of the scale matrix in a multivariate skew-normal distribution. We propose a statistic for testing whether the skewness vector is proportional to one of the eigenvectors of the scale matrix based on the likelihood ratio. Under the alternative, the likelihood is maximized numerically with two different ways of parametrization for the scale matrix: Modified Cholesky Decomposition (MCD) and Givens Angle. We conduct a simulation study and show that the statistic obtained using Givens Angle parametrization performs well and is more reliable than that obtained using MCD.