Robust Statistics, 2nd Edition

3Measuring Robustness

In order to measure the effect of different locations of an outlier on an estimate, consider adding to a sample an extra data point that is allowed to range on the whole real line. We define the sensitivity curve of the estimator for the sample as the difference

3.1

as a function of the location of the outlier.

For purposes of plotting and comparing sensitivity curves across sample sizes, it is convenient to use standardized sensitivity curves, obtained by multiplying (3.1) by (see also Section 3.1). To make our examples clearer, we use a “sample” formed from standard normal distribution quantiles, instead of a random one. Figure 3.1 plots:

• the standardized sensitivity curves of the median
• the 10% trimmed mean
• the 10% Winsorized mean
• the Huber M‐estimator with ...