Minimum Average Variance Estimation with group Lasso for the multivariate response Central Mean Subspace

Abstract

The Minimum Average Variance Estimation (MAVE) method and its variants have proven to be effective approaches to the dimension reduction problems. However, as far as we know, using MAVE to estimate the Central Mean Subspace (CMS) with multivariate response receives little attention. This paper proposes a weighted version of MAVE for the CMS with multivariate response. The proposed weighted MAVE method takes account of the correlations among the responses and is a natural extension of the original MAVE method that only targets the univariate response CMS. The algorithm to implement the weighted MAVE method is provided. Asymptotic distribution of the MAVE estimator under the multivariate response setting is also derived. In addition, for the goal of variable selection, we propose incorporating the adaptive group lasso regularization method to induce sparse solutions. We show that the resulting sparse estimator can be calculated in a computationally efficient manner, and the associated BIC-type criterion can consistently select relevant variables. Experimental simulations and a real data analysis demonstrate the effectiveness and usefulness of the proposed methods.