Robust estimation in very small samples

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 ≥ 3, whereas for scale n ≥ 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.