High-dimensional Covariance Estimation by Pairwise Likelihood Truncation

Pairwise likelihood offers a useful approximation to the full likelihood function for covariance estimation in high-dimensional context. It simplifies high-dimensional dependencies by combining marginal bivariate likelihood objects, thereby making estimation more manageable. In certain models, including the Gaussian model, both pairwise and full likelihoods are known to be maximized by the same parameter values, thus retaining optimal statistical efficiency, when the number of variables is fixed. Leveraging this insight, we introduce the estimation of sparse high-dimensional covariance matrices by maximizing a truncated version of the pairwise likelihood function, which focuses on pairwise terms corresponding to nonzero covariance elements. To achieve a meaningful truncation, we propose to minimize the discrepancy between pairwise and full likelihood scores plus an L1-penalty discouraging the inclusion of uninformative terms. Differently from other regularization approaches, our method selects whole pairwise likelihood objects rather than individual covariance parameters, thus retaining the inherent unbiasedness of the pairwise likelihood estimating equations. This selection procedure is shown to have the selection consistency property as the covariance dimension increases exponentially fast. As a result, the implied pairwise likelihood estimator is consistent and converges to the oracle maximum likelihood estimator that assumes knowledge of nonzero covariance entries.