Asymptotic behaviour of penalized robust estimators in logistic regression when dimension increases

Abstract

In the framework of logistic regression in order to obtain sparse modelsand automatic variable selection, penalized M-estimators that boundthe deviance have been previously studied for fixed dimension. Inthis chapter, we consider a wide class of M-estimators that involvessome well-known robust proposals and study their asymptotic behaviour when the covariates dimension grows to infinity with the sample size.Among other results, we obtain consistency, rates of convergence, and weexplore the oracle properties of the regularized M-estimators, for penaltyfunctions of different nature. Specifically, under suitable conditions, weprove that, with probability tending to 1, these estimators only selectvariables corresponding to non-null true coefficients, and we derive theirasymptotic distribution.