Robust Methods in Regression Analysis – Theory and Application

Regression Analysis is an important statistical tool for many applications. The most
frequently used approach to Regression Analysis is the method of Ordinary Least Squares.
But this method is vulnerable to outliers; even a single outlier can spoil the estimation
completely. How can this vulnerability be described by theoretical concepts and are there alternatives? This thesis gives an overview over concepts and alternative approaches.
The three fundamental approaches to Robustness (qualitative-, infinitesimal- and quantitative Robustness) are introduced in this thesis and are applied to different estimators. The estimators under study are measures of location, scale and regression. The Robustness approaches are important for the theoretical judgement of certain estimators but as well for the development of alternatives to classical estimators. This thesis focuses on the (Robustness-) performance of estimators if outliers occur within the data set. Measures of location and scale provide necessary steppingstones into the topic of Regression Analysis. In particular the median and trimming approaches are found to produce very robust results.
These results are used in Regression Analysis to find alternatives to the method of Ordinary Least Squares. Its vulnerability can be overcome by applying the methods of Least Median of Squares or Least Trimmed Squares. Different outlier diagnostic tools are introduced to improve the poor efficiency of these Regression Techniques. Furthermore, this thesis delivers a simulation of some Regression Techniques on different situations in Regression Analysis.
This simulation focuses in particular on changes in regression estimates if outliers occur in the data.
Theoretically derived results as well as the results of the simulation lead to the
recommendation of the method of Reweighted Least Squares. Applying this method
frequently on problems of Regression Analysis provides outlier resistant and efficient
estimates.