Resumen de Robust and sparse estimation of large precision matrices
The thesis considers the estimation of sparse precision matrices in the high-dimensional setting. First, we introduce an integrated approach to estimate undirected graphs and to perform model selection in high-dimensional Gaussian Graphical Models (GGMs). The approach is based on a parametrization of the inverse covariance matrix in terms of the prediction errors of the best linear predictor of each node in the graph. We exploit the relationship between partial correlation coefficients and the distribution of the prediction errors to propose a novel forward-backward algorithm for detecting pairs of variables having nonzero partial correlations among a large number of random variables based on i.i.d. samples. Then, we are able to establish asymptotic properties under mild conditions. Finally, numerical studies through simulation and real data examples provide evidence of the practical advantage of the procedure, where the proposed approach outperforms state-of-the-art methods such as the Graphical lasso and CLIME under different settings.
Furthermore, we study the problem of robust estimation of GGMs in the high-dimensional setting when the data may contain outlying observations. We propose a robust precision matrix estimator under the cellwise contamination mechanism that is robust against structural bivariate outliers. This framework exploits robust pairwise weighted correlation coefficient estimates, where the weights are computed by the Mahalanobis distance with respect to an affine equivariant robust correlation coefficient estimator. We show that the convergence rate of the proposed estimator is the same as the correlation coefficient used to compute the Mahalanobis distance. We conduct numerical simulation under different contamination settings to compare the graph recovery performance of different robust estimators. The proposed method is then applied to the classification of tumors using gene expression data. We show that our procedure can effectively recover the true graph under cellwise data contamination.
