Quasi-Likelihood and/or Robust Estimation in High Dimensions

Abstract

We consider the theory for the high-dimensional generalized linear model with the Lasso. After a short review on theoretical results in literature, we present an extension of the oracle results to the case of quasi-likelihood loss. We prove bounds for the prediction error and ${\ell }_1$-error. The results are derived under fourth moment conditions on the error distribution. The case of robust loss is also given. We moreover show that under an irrepresentable condition, the ${\ell }_1$-penalized quasi-likelihood estimator has no false positives.

Published in at http://dx.doi.org/10.1214/12-STS397 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (7)

• High-dimensional generalized linear models and the lasso
• Sparsity oracle inequalities for the Lasso
• L1-Penalization for Mixture Regression Models
• Aggregation for Gaussian regression
• Estimation for High-Dimensional Linear Mixed-Effects Models Using ${\ell }_1$-Penalization
• The Dantzig selector and sparsity oracle inequalities
• Sup-norm convergence rate and sign concentration property of Lasso and Dantzig estimators

Cited by in corpus (1)

• Introduction to the Special Issue on Sparsity and Regularization Methods