Robust estimation and variable selection for sparse generalized linear models.

• Department of Mathematics, University of Trento, Trento, Italy [claudio.agostinelli@unitn.it]

• Instituto de Cálculo, University of Buenos Aires and CONICET, Buenos Aires, Argentina [mvaldora@dm.uba.ar]

Keywords: Generalized linear models – Variable Selection – Robustness – High dimension

1 Introduction

A generalized linear model (GLM) is called sparse when only a small proportion of the regression parameters are different from zero. Estimating the model parameters and identifying which ones are zero leads to a variable selection criterion. This problem has been widely studied in recent years and a well-known approach to addressing it is to introduce a penalization term.

Classical penalized estimators are highly sensitive to outliers, particularly when these outliers correspond to high-leverage observations.

2 Robust estimation and variable selection in GLMs

We propose a method for robust estimation and variable selection in generalized linear models. These estimators are defined as penalized MT-estimators. MT-estimators were proposed in Valdora and Yohai [2014] and further studied in Agostinelli et al. [2019]. In Valdora and Agostinelli [2023] penalized elastic net MT-estimators were proposed.

In this work, we study penalized MT-estimators, considering different types of penalties and we study their variable selection properties.

We prove that, under suitable assumptions, the proposed estimators are consistent and possess the oracle property.

We evaluate the finite sample performance of this method for Poisson distribution and well known penalization functions through Monte Carlo simulations that consider different models and contaminations, as well as an empirical application using a real dataset.