Journal
COMPUTATIONAL STATISTICS & DATA ANALYSIS
Volume 40, Issue 4, Pages 741-758Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/S0167-9473(02)00078-6
Keywords
breakdown value; stylized empirical influence function; small data sets; three-dimensional empirical influence function
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In experimental science measurements are typically repeated only a few times, yielding a sample size n of the order of 3 to 8. One then wants to summarize the measurements by a central value and measure their variability, i.e. estimate location and scale. These estimates should preferably be robust against outliers, as reflected by their small-sample breakdown value. The estimator's stylized empirical influence function should be smooth, monotone increasing for location, and decreasing-increasing for scale. It turns out that location can be estimated robustly for n greater than or equal to 3, whereas for scale n greater than or equal to 4 is needed. Several well-known robust estimators are studied for small n, yielding some surprising results. For instance, the Hodges-Lehmann estimator equals the average when n=4. Also location M-estimators with auxiliary scale are studied, addressing issues like the difference between one-step and fully iterated M-estimators. Simultaneous M-estimators of location and scale (`Huber's Proposal 2') are considered as well, and it turns out that their lack of robustness is already noticeable for such small samples. Recommendations are given as to which estimators to use. (C) 2002 Elsevier Science B.V. All rights reserved.
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