On the existence and uniqueness of estimates in robust and heteroscedastic regression models

The thesis studies redescending M-estimators for the ordinary linear regression model, and maximum likelihood estimators for heteroscedastic regression models. In general, redescending M-estimators do not yield unique estimates of the model parameters, and the thesis shows that the difficulties associated with this have not always been fully appreciated in the literature. This motivates the development of an approach whereby unique redescending M-estimates can be reliably obtained. This is achieved by embedding the linear model within a multivariate t location-scatter framework, which is known in the literature for its desirable uniqueness properties. M-estimates derived from the conditional t distribution are also considered, but it is shown that the resulting objective function is intrinsically multimodal, with modes of infinity. The nonregularity result for the conditional t model is found to have implications for heteroscedastic regression models. Two classes of commonly proposed models are considered. The first is found to yield an unbounded likelihood at points corresponding to nonreplicated observations, whilst for the second a much stronger linear independence condition is obtained for the likelihood to be unbounded. The thesis concludes with a discussion of efficient methods for testing analytical conditions arising from the preceding studies. Keywords: conditional distribution, heteroscedastic regression, mean-variance relationship, multimodality, multivariate t distribution, redescending M-estimator, robust regression, singularity, uniqueness