On Robustness in some extended regression models

Abstract

Generalized Linear Models extends classical regression models to non-normal response variables and allows a non-linear relation between the mean of the responses and the predictors. In addition, when the responses are correlated or show overdispersion, one can add a linear combination of random components to the linear predictor. The resulting models are known as Generalized Linear Mixed Models. Traditional estimation methods in these classes of models rely on distributional assumptions about the random components, as well as the implicit assumption that the explanatory variables are uncorrelated with the error term. In Chapters 2 and 3 we investigate, using the Change-of-Variance Function, the behavior of the asymptotic variance-covariance matrix of the class of M-estimators when the distribution of the random components is slightly contaminated. In Chapter 4 we study a different concept of robustness for classical models that contain explanatory variables correlated with the error term. For these models we propose an instrumental variables estimator and study its robustness by means of its Influence Function.

We extend the definitions of Change-of-Variance Function to Generalized Linear Models and Generalized Linear Mixed Models. We use them to analyze in detail the sensitivity of the asymptotic variance of the maximum likelihood estimator. For the first class of models, we found that, in general, a contamination of the distribution can seriously affect the asymptotic variance of the estimators. For the second class, we focus on the Poisson-Gamma model and two mixed-effects Binomial models. We found that the effect of a contamination in the mixing distribution on the asymptotic variance of the maximum likelihood estimator remain bounded for both models. A simulation study was performed in all cases to illustrate the relevance of our results.

Finally, we propose a robust instrumental variables estimator based on high breakdown point S-estimators of location and scatter. The resulting estimator has bounded Influence Function and satisfies the usual asymptotic properties for suitable choices of the S-estimator used. We also derive an estimate for the asymptotic covariance matrix of our estimator which is robust against outliers and leverage points. We illustrate our results using a real data example.