| A logical starting point for a discussion of Pareto and Pareto-like distributions is Vilfredo Pareto's Economics textbook published in Rome in 1897. In it he observed that the number of persons in a population whose incomes exceed a certain level is often well approximated by a hyperbolic function. Besides this application, such models arise in actuarial science, economics, telecommunications, and reliability, for example, as well as in the modeling of upper observations in samples from distributions with regularly varying tails.; In this dissertation, robust estimation of the tail index parameter of a Pareto model is treated. A systematic study of various existing methods popular in the literature is carried out, applying two generally competing criteria: efficiency and robustness. Specifically, asymptotic relative efficiencies with respect to the maximum likelihood estimator (MLE), and breakdown points are examined. In particular, maximum likelihood, method of moments, least squares, quantile type and trimmed mean type estimators are considered.; The established results demonstrate that the method of moments and the least squares estimators are deficient with respect to both criteria and should become disfavored, while the trimmed mean type estimators tend to dominate the other abovementioned competitors to the MLE.; Further, and very significantly, new tail index estimators of “generalized median statistic” type are introduced and studied. It turns out that these estimators dominate all the other estimators under consideration. The superiority of the generalized median statistics may be explained by the following principle: “Smoothing” followed by “medianing” produces estimators which exhibit a very favorable trade-off between efficiency and robustness. The results of the dissertation provide a basis for revision of some prevailing viewpoints and methods. |
