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

TL;DR

This paper investigates the asymptotic properties of penalized robust estimators in high-dimensional logistic regression, establishing consistency, variable selection, and distributional results as both sample size and dimension grow.

Contribution

It extends existing fixed-dimension results to high-dimensional settings, providing new asymptotic theory for penalized M-estimators in logistic regression.

Findings

Establishes consistency of estimators as dimension increases

Proves variable selection consistency with high probability

Derives asymptotic distribution of estimators in high dimensions

Abstract

Penalized $M-$estimators for logistic regression models have been previously study for fixed dimension in order to obtain sparse statistical models and automatic variable selection. In this paper, we derive asymptotic results for penalized $M-$estimators when the dimension $p$ grows to infinity with the sample size $n$. Specifically, we obtain consistency and rates of convergence results, for some choices of the penalty function. Moreover, we prove that these estimators consistently select variables with probability tending to 1 and derive their asymptotic distribution.