Estimation of Multivariate Dependence Structures via Constrained Maximum Likelihood

Estimating high-dimensional dependence structures in models of multivariate datasets is an ongoing challenge. Copulas provide a powerful and intuitive way to model dependence structure in the joint distribution of disparate types of variables. Here, we propose an estimation method for Gaussian copula parameters based on the maximum likelihood estimate of a covariance matrix that includes shrinkage and where all of the diagonal elements are restricted to be equal to 1. We show that this estimation problem can be solved using a numerical solution that optimizes the problem in a block coordinate descent fashion. We illustrate the advantage of our proposed scheme in providing an efficient estimate of sparse Gaussian copula covariance parameters using a simulation study. The sparse estimate was obtained by regularizing the constrained problem using either the least absolute shrinkage and selection operator (LASSO) or the adaptive LASSO penalty, applied to either the covariance matrix or the inverse covariance (precision) matrix. Simulation results indicate that our method outperforms conventional estimates of sparse Gaussian copula covariance parameters. We demonstrate the proposed method for modelling dependence structures through an analysis of multivariate groundfish abundance data obtained from annual bottom trawl surveys in the northeast Pacific from 2014 to 2018. Supplementary materials accompanying this paper appear on-line.