Small sample performance of robust estimators of tail parameters for Pareto and exponential models

Robust estimation of tail index parameters is treated for (equivalent) two-parameter Pareto and exponential models. These distributions arise as parametric models in actuarial science, economics, telecommunications, and reliability, for example, as well as in semiparametric modeling of upper observations in samples from distributions which are regularly varying or in the domain of attraction of extreme value distributions. In a recent previous paper, new estimators of "generalized median" (GM) type were introduced and shown to provide more favorable trade-offs between efficiency and robustness than several well-established estimators, including those corresponding to methods of maximum likelihood, trimming, and quantiles. Here we establish-via simulation-that the superiority of the GM type estimators remains valid even for small sample sizes n = 10 and 25. To bridge between "small" and "large" sample sizes, we also include the cases n = 50 and 100. Further, we arrive at guidelines for selection of a particular GM estimator in practice, depending upon the sample size, upon whether protection is desired against upper outliers only, or against both upper and lower outliers, and upon whether the level of possible contamination by outliers is high or low. Comparisons of estimators are made on the basis of relative efficiency with respect to the maximum likelihood estimator, breakdown points, and premium-protection plots.