Analysis of Robust Quasi-deviances for Generalized Linear Models

Abstract

Generalized linear models (McCullagh and Nelder 1989) are a popular technique for modeling a large variety of continuous and discrete data. They assume that the response variables Y_i , for i = 1, . . . , n, come from a distribution belonging to the exponential family, such that E[Y_i ] = µ_i and V[Y_i ] = V (µ_i ), and that η_i = g(µ_i ) = x_i^Tβ, where β ∈ IR ^p is the vector of parameters, x_i ∈ IR ^p, and g(.) is the link function.

The non-robustness of the maximum likelihood and the maximum quasi-likelihood estimators has been studied extensively in the literature. For model selection, the classical analysis-of-deviance approach shares the same bad robustness properties. To cope with this, Cantoni and Ronchetti (2001) propose a robust approach based on robust quasi-deviance functions for estimation and variable selection. We refer to that paper for a deeper discussion and the review of the literature.