On Robustness in some extended regression models

Generalized Linear Models extends classical regression models tonon-normal response variables and allows a non-linear relationbetween the mean of the responses and the predictors. In addition,when the responses are correlated or show overdispersion, one canadd a linear combination of random components to the linearpredictor. The resulting models are known as Generalized LinearMixed Models. Traditional estimation methods in these classes ofmodels rely on distributional assumptions about the randomcomponents, as well as the implicit assumption that theexplanatory variables are uncorrelated with the error term. InChapters 2 and 3 we investigate, using the Change-of-VarianceFunction, the behavior of the asymptotic variance-covariancematrix of the class of M-estimators when the distribution of therandom components is slightly contaminated. In Chapter 4 we studya different concept of robustness for classical models thatcontain explanatory variables correlated with the error term. Forthese models we propose an instrumental variables estimator andstudy its robustness by means of its Influence Function.We extend the definitions of Change-of-Variance Function toGeneralized Linear Models and Generalized Linear Mixed Models. Weuse them to analyze in detail the sensitivity of the asymptoticvariance of the maximum likelihood estimator. For the first classof models, we found that, in general, a contamination of thedistribution can seriously affect the asymptotic variance of theestimators. For the second class, we focus on the Poisson-Gammamodel and two mixed-effects Binomial models. We found that theeffect of a contamination in the mixing distribution on theasymptotic variance of the maximum likelihood estimator remainbounded for both models. A simulation study was performed in allcases to illustrate the relevance of our results.Finally, we propose a robust instrumental variables estimatorbased on high breakdown point S-estimators of location andscatter. The resulting estimator has bounded Influence Functionand satisfies the usual asymptotic properties for suitable choicesof the S-estimator used. We also derive an estimate for theasymptotic covariance matrix of our estimator which is robustagainst outliers and leverage points. We illustrate our resultsusing a real data example.