Robust Inference for Generalized Linear Models

Abstract

By starting from a natural class of robust estimators for generalized linear models based on the notion of quasi-likelihood, we define robust deviances that can be used for stepwise model selection as in the classical framework.We derive the asymptotic distribution of tests based on robust deviances and we investigate the stability of their asymptotic level under contamination. The binomial and Poisson models are treated in detail. Two applications to real data and a sensitivity analysis show that the inference obtained by means of the new techniques is more reliable than that obtained by classical estimation and testing procedures.

A robust approach for skewed and heavy-tailed outcomes in the analysis of health care expenditures

“…Because the GLM technique is based on maximum likelihood or quasi-likelihood, it is very sensitive to spurious observations 2 . Cantoni and Ronchetti (2001) developed robust versions of estimators and tests for GLM in the case of binomial and Poisson models.…”

“…logistic), as treated in detail in Cantoni and Ronchetti (2001). An alternative approach would consider specific distributions that model directly the mass at zero, either via the likelihood of an hurdle model or via a zero-inflated distribution 4 .…”

“…To address robustness (in the sense of local stability, as measured by the influence function), Cantoni and Ronchetti (2001) suggested to estimate the parameter β via M-estimation (Huber, 1981), that is through a set of estimating equations of the form n i=1 Ψ(y i , β, ν) = 0. The idea is to build upon the classical estimating equations…”

“…Moreover, under quite general conditions, it can be shown (see Cantoni and Ronchetti, 2001) that the asymptotic distribution of √ n(β − β), where β is the solution of (5), is normal with expectation 0 and variance equal to…”

“…This can be done for several model distributions including binomial and Poisson (see Cantoni and Ronchetti, 2001) and Gamma (see Appendix A). For other distributions, these terms can be at least approximated numerically.…”