Robust estimation of high-dimensional covariance and precision matrices.

High-dimensional data are often most plausibly generated from distributions with complex structure and leptokurtosis in some or all components. Covariance and precision matrices provide a useful summary of such structure, yet the performance of popular matrix estimators typically hinges upon a sub-Gaussianity assumption. This paper presents robust matrix estimators whose performance is guaranteed for a much richer class of distributions. The proposed estimators, under a bounded fourth moment assumption, achieve the same minimax convergence rates as do existing methods under a sub-Gaussianity assumption. Consistency of the proposed estimators is also established under the weak assumption of bounded 2 + ε moments for ε ∈ (0, 2). The associated convergence rates depend on ε.

A Predictive Based Regression Algorithm for Gene Network Selection.

Gene selection has become a common task in most gene expression studies. The objective of such research is often to identify the smallest possible set of genes that can still achieve good predictive performance. To do so, many of the recently proposed classification methods require some form of dimension-reduction of the problem which finally provide a single model as an output and, in most cases, rely on the likelihood function in order to achieve variable selection. We propose a new prediction-based objective function that can be tailored to the requirements of practitioners and can be used to assess and interpret a given problem. Based on cross-validation techniques and the idea of importance sampling, our proposal scans low-dimensional models under the assumption of sparsity and, for each of them, estimates their objective function to assess their predictive power in order to select. Two applications on cancer data sets and a simulation study show that the proposal compares favorably with competing alternatives such as, for example, Elastic Net and Support Vector Machine. Indeed, the proposed method not only selects smaller models for better, or at least comparable, classification errors but also provides a set of selected models instead of a single one, allowing to construct a network of possible models for a target prediction accuracy level.