Strategies for mean and modal multivariate local regression

Local polynomial fitting for univariate data has been widely studied and discussed, but up until now the multivariate equivalent has often been deemed impractical, due to the so-called "curse of dimensionality". Here, rather than discounting it completely, density is used as a threshold to determine where over a data range reliable multivariate smoothing is possible, whilst accepting that in large areas it is not. Further, the challenging issue of multivariate bandwidth selection, which is known to be affected detrimentally by sparse data which inevitably arise in higher dimensions, is considered. In an effort to alleviate this problem, two adaptations to generalized cross-validation are implemented, and a simulation study is presented to support the proposed method. It is also discussed how the density threshold and the adapted generalized cross-validation technique introduced herein work neatly together. Whilst this is the major focus of this thesis, modal regression via mean shift is discussed as an alternative multivariate regression technique. In a slightly different vein, bandwidth selection for univariate kernel density estimation is also examined, and a different technique is proposed for a density with a multimodal distribution. This is supported by a simulation study and its relevance in modal regression is also discussed.