High-Dimensional Precision Matrix Estimation Under Total Positivity

We consider the problem of estimating high-dimensional precision matrices for multivariate totally positive of order two (MTP2) distributions. Such distribution is of interest in financial econometrics, where the distribution of stock price daily change is usually MTP2. A large body of methods have been proposed for estimating precision matrices in general high-dimensional distributions. However, they either require strong and non-interpretable conditions or require many tuning parameters for structure recovery. We propose a regularized maximum likelihood-based estimator for MTP2 distributions and establish graphical structure recovery guarantees (sparsistency) of the proposed estimator in the high-dimensional setting under a set of interpretable conditions. Our theoretical and simulation analysis shows that these conditions are usually naturally satisfied for MTP2 distributions. We also prove that the sparsistency is robust with respect to the choice of tuning parameter and analyse the performance of the proposed estimator on simulated and stock pricing data. As a corollary we generalize our method to estimating precision matrices in the general high-dimensional distributions and prove that the maximum weight spanning forest of the sample correlation matrix is also consistent in discovering a subset of true positive edges for high-dimensional MTP2 distributions.