Manuel Koller: Robust Statistics: Tests for Robust Linear Regression

Adviser: Werner Stahel

March 2008

Abstract:

Analyzing data using statistical methods means to break reality down to
a mathematical framework, a model. Often this model is based on strong
assumptions, for example normally distributed data. Classical statistics
provides methods that fit the chosen model perfectly. But in reality the
model assumptions usually hold only approximately. Anomalies and untrue
assumptions might render the statistical analysis useless.

Robust statistics aims for methods that are based on weaker assumptions
and thus allow small deviations from the classical model.  However,
robust statistics is not restricted to the use of robust estimation
methods alone. It also extends to methods used to draw inference. In the
past, there has not been much research focused on robust tests.

In this thesis we study the quality of inference performed by of two
state-of-the-art robust regression procedures. We then propose a design
adapted scale estimator and use it as part of a new robust regression
estimator, the MMD-estimator. This new estimator improves the quality of
robust tests considerably.

A simulation study is performed to compare the performance of the
mentioned regression procedures in combination with various covariance
matrix estimators. We found large differences between the tested
methods. Some methods were able to approximately reach the desired level
in the corresponding tests for most tested scenarios while others
produced estimates that were only useful in specific high sample
settings. We infer that the covariance matrix estimator needs to be
carefully selected for every new scenario.

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