Robust Alternatives to Least SquaresWerner

Least Squares estimation in regression models is well known to be optimal if the random errors are normally distributed. In the presence of longer-tailed errors or outliers, there are better alternatives, called robust estimators. Two robustness measures are based on the idea of gross errors { arbitrary observations added to a given sample. The gross error sensitivity measures the maximal eeect of a single observation, and the breakdown point characterizes the fraction of gross errors that leads to arbitrarily large eeects on the estimate. M-estimators are a exible class of estimators suitable for obtaining robust procedures with desired properties. In the regression context, they show only limited and often unsatisfactory robustness. On the other hand, desired robustness properties can only be achieved at the cost of heavy computation. These concepts are explained on the basis of two examples from environmental sciences and molecular spectroscopy. In the latter, robust methods were essential for nding a useful solution.