Local Smoothing Methods for the Analysis of Multivariate Complex Data Structures

Local smoothing methods are a widely used tool in the context of
nonparametric regression. The essential idea is to perform a linear
or polynomial regression locally in a neighborhood of the target
point. This method is generalized in two ways. Firstly, the
polynomials are substituted by arbitrary smooth basis functions,
and secondly, the estimating methodology, which is based on the
least squares method, is modified in a suitable way. It appears
that the first concept is useful for bias reduction, while the
second one is interesting for robustifcation against outliers in
the predictors. As by-products some interesting relations to other
mathematical and statistical topics are unveiled, concerning in
particular the theorems from Taylor and Horvitz-Thompson. In the
further course of the thesis the interest turns to some particular
problems which have not been a domain of local methods so far. It
turns out that local smoothing methods, suitably combined, are
useful for the online monitoring of time series in order to detect
sudden breaks or jumps. Finally, the restriction of modelling only
functional data is abandoned and a new approach to calculate
principal curves, i.e. smooth curves which pass through the
``middle'' of a multidimensional, possibly multiply branched, data
cloud, is developed.