Estimation of the Tail Index Using Semiparametric Regression

• Activity Number: 106 - Empirical Transforms, Saddlepoint Approximations, and Risk Assessment with Some Applications
• Type: Topic Contributed
• Date/Time: Monday, July 31, 2017 : 8:30 AM to 10:20 AM
• Sponsor: Section on Risk Analysis
• Author(s): Emanuele Taufer* and Mofei Jia and Maria Michela Dickson
• Companies: University of Trento and Xi'an Jiaotong-Liverpool University and University of Trento
• Keywords: Heavy-tailed distributions ; regular variation ; empirical characteristic function ; Zipf's law

Abstract

Consider a distribution F with regularly varying tails of index ?. An estimation strategy for ?, exploiting the relation between the behavior of the tail at infinity and that of the characteristic function near the origin is proposed. A semi-parametric regression model does the job: a non-parametric component controls the bias and a parametric one produces the actual estimate. Implementation of the estimation strategy is quite simple as it can rely on standard software packages for generalized additive models. A generalized cross validation procedure is suggested in order to handle the bias-variance trade-off. The theoretical properties of the proposed procedure, consistency and asymptotic normality, are derived. Simulations show the excellent performance of this estimator in a wide range of cases. An application to data sets on city sizes, facing the debated issue of distinguishing Pareto-type tails from Log-normal tails, illustrates how the proposed method works in practice.