Dual Model Robust Regression

In typical normal theory regression, the assumption ofhomogeneity of variances is often not appropriate.Instead of treating the variances as a nuisance andtransforming away the heterogeneity, the structure ofthe variances may be of interest and it is desirable tomodel the variances. Aitkin (1987) proposes aparametric dual model in which a log lineardependence of the variances on a set of explanatoryvariables is assumed. Aitkin's parametric approach isan iterative one providing estimates for the parametersin the mean and variance models through jointmaximum likelihood. Estimation of the mean andvariance parameters are interrelatedas the responsesin the variance model are the squared residuals fromthe fit to the means model. When one or both of themodels (the mean or variance model) aremisspecified, parametric dual modeling can lead tofaulty inferences. An alternative to parametric dualmodeling is to let the data completely determine theform of the true underlying mean and variancefunctions (nonparametric dual modeling). However,nonparametric techniques often result in estimateswhich are characterized by high variability and theyignore important knowledge that the user may haveregarding the process. Mays and Birch (1996) havedemonstrated an effective semiparametric method inthe one regressor, single-model regression settingwhich is a "hybrid" of parametric and nonparametricfits. Using their techniques, we develop a dualmodeling approach which is robust to misspecificationin either or both of the two models. Examples will bepresented to illustrate the new technique, termed hereas Dual Model Robust Regression.