Robust estimation of the Pareto tail index: a Monte Carlo analysis

Abstract

The Pareto distribution is often used in many areas of economics to model the right tail of heavy-tailed distributions. However, the standard method of estimating the shape parameter (the Pareto tail index) of this distribution—the maximum likelihood estimator (MLE), also known as the Hill estimator—is non-robust, in the sense that it is very sensitive to extreme observations, data contamination or model deviation. In recent years, a number of robust estimators for the Pareto tail index have been proposed, which correct the deficiency of the MLE. However, little is known about the performance of these estimators in small-sample setting, which often occurs in practice. This paper investigates the small-sample properties of the most popular robust estimators for the Pareto tail index, including the optimal B-robust estimator (Victoria-Feser and Ronchetti in Can J Stat 22:247–258, 1994), the weighted maximum likelihood estimator (Dupuis and Victoria-Feser in Can J Stat 34:639–658, 2006), the generalized median estimator (Brazauskas and Serfling in Extremes 3:231–249, 2001), the partial density component estimator (Vandewalle et al. in Comput Stat Data Anal 51:6252–6268, 2007), and the probability integral transform statistic estimator (PITSE) (Finkelstein et al. in N Am Actuar J 10:1–10, 2006). Monte Carlo simulations show that the PITSE offers the desired compromise between ease of use and power to protect against outliers in the small-sample setting.