Adaptive robust variable selection

Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weighted $L_1$-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the $L_1$-penalty. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, we investigate the model selection oracle property and establish the asymptotic normality of the WR-Lasso. We show that only mild conditions on the model error distribution are needed. Our theoretical results also reveal that adaptive choice of the weight vector is essential for the WR-Lasso to enjoy these nice asymptotic properties. To make the WR-Lasso practically feasible, we propose a two-step procedure, called adaptive robust Lasso (AR-Lasso), in which the weight vector in the second step is constructed based on the $L_1$-penalized quantile regression estimate from the first step. This two-step procedure is justified theoretically to possess the oracle property and the asymptotic normality. Numerical studies demonstrate the favorable finite-sample performance of the AR-Lasso.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1191 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Variable selection using MM algorithms

Variable selection is fundamental to high-dimensional statistical modeling. Many variable selection techniques may be implemented by maximum penalized likelihood using various penalty functions. Optimizing the penalized likelihood function is often challenging because it may be nondifferentiable and/or nonconcave. This article proposes a new class of algorithms for finding a maximizer of the penalized likelihood for a broad class of penalty functions. These algorithms operate by perturbing the penalty function slightly to render it differentiable, then optimizing this differentiable function using a minorize-maximize (MM) algorithm. MM algorithms are useful extensions of the well-known class of EM algorithms, a fact that allows us to analyze the local and global convergence of the proposed algorithm using some of the techniques employed for EM algorithms. In particular, we prove that when our MM algorithms converge, they must converge to a desirable point; we also discuss conditions under which this convergence may be guaranteed. We exploit the Newton-Raphson-like aspect of these algorithms to propose a sandwich estimator for the standard errors of the estimators. Our method performs well in numerical tests.Comment: Published at http://dx.doi.org/10.1214/009053605000000200 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Shrinkage and Variable Selection by Polytopes

Constrained estimators that enforce variable selection and grouping of highly correlated data have been shown to be successful in finding sparse representations and obtaining good performance in prediction. We consider polytopes as a general class of compact and convex constraint regions. Well established procedures like LASSO (Tibshirani, 1996) or OSCAR (Bondell and Reich, 2008) are shown to be based on specific subclasses of polytopes. The general framework of polytopes can be used to investigate the geometric structure that underlies these procedures. Moreover, we propose a specifically designed class of polytopes that enforces variable selection and grouping. Simulation studies and an application illustrate the usefulness of the proposed method