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

Abstract

In the framework of logistic regression in order to obtain sparse models and automatic variable selection, penalized M-estimators that bound the deviance have been previously studied for fixed dimension. In this chapter, we consider a wide class of M-estimators that involves some 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 we explore the oracle properties of the regularized M-estimators, for penalty functions of different nature. Specifically, under suitable conditions, we prove that, with probability tending to 1, these estimators only select variables corresponding to non-null true coefficients, and we derive their asymptotic distribution.