Asymptotic Results for Model Robust Regression

Since the mid 1980's many statisticians have studied methods for combining parametric andnonparametric esimates to improve the quality of fits in a regression problem. Notably in 1987,Einsporn and Birch proposed the Model Robust Regression estimate (MRR1) in which estimatesof the parametric function, f, and the nonparametric function, g, were combined in astraightforward fashion via the use of a mixing parameter, l. This technique was studiedextensively at small samples and was shown to be quite effective at modeling various unusualfunctions. In 1995, Mays and Birch developed the MRR2 estimate as an alternative to MRR1.This model involved first forming the parametric fit to the data, and then adding in an estimate ofg according to the lack of fit demonstrated by the error terms. Using small samples, theyillustrated the superiority of MRR2 to MRR1 in most situations. In this dissertation we havedeveloped asymptotic convergence rates for both MRR1 and MRR2 in OLS and GLS (maximumlikelihood) settings. In many of these settings, it is demonstrated that the user of MRR1 orMRR2 achieves the best convergence rates available regardless of whether or not the model isproperly specified. This is the "Golden Result of Model Robust Regression". It turns out that theselection of the mixing parameter is paramount in determining whether or not this result isattained.