A Review of More than One Hundred Pareto-Tail Index Estimators

Abstract

Heavy-tailed distributions are often encountered in economics, finance, biology, telecommunications, geology, etc. The heaviness of a tail is measured by a tail index. Numerous methods for tail index estimation have been proposed. This paper reviews more than one hundred Pareto (and equivalent) tail index estimators. It focuses on univariate estimators for non-truncated data. We discuss the basic features of these estimators and provide their analytical expressions. As samples from heavy-tailed distributions are often analysed by researchers from various sciences, the paper provides nontechnical explanations of the methods, so as to be understood by researchers with intermediate skills in statistics. We also discuss the strengths and weaknesses of the estimators, if known. The main focus of the paper is semi-parametric estimators; however, a number of parametric estimators under-represented in previous reviews are also discussed. The paper can be viewed as a catalog or a reference work on Pareto-tail index estimators. A Monte-Carlo comparison of more than 90 estimators is presented.

Keywords

• heavy tails
• pareto distribution
• tail index
• review