Data Analysis and Related Applications, Volume 1

17The Minimum Renyi’s Pseudodistance Estimators for Generalized Linear Models

Minimum Renyi’s pseudodistance (RP) estimators have good robustness properties without a significant loss of efficiency for linear regression models (LRM). The main purpose of this chapter is to extend these minimum RP estimators to generalized linear models (GLM), using some results previously obtained by Castilla et al. (2021) in relation to independent and non-identically distributed observations in LRM. We theoretically derive asymptotic properties of the proposed estimators and examine the performance of the estimators in Poisson regression models through a simulation study, focusing on the robustness properties of the estimators. We finally test the proposed methods in a real dataset related to the treatment of epilepsy, illustrating the outperformance of the robust minimum RP estimators when there are outlier observations.

17.1. Introduction

Generalized linear models (GLMs) were first introduced by Nelder and Wedderburn (1972) and later widely by McCullagh and Nelder (1983). The GLMs represent a natural extension of the standard linear regression model (LRM), which encloses a large variety of response variable distributions, including distributions of counts, binary or positive values. The regression model is defined in terms of a set of independent response variables, Y₁, ..., Y_n, following a distribution from the general exponential family. That is, the density function of each response ...