Estimation of the tail index of Pareto-type distributions using regularisation

TL;DR

This paper presents new reduced-bias estimators for the tail index of Pareto-type distributions, utilizing regularised weighted least squares with exponential regression, showing improved bias and mean squared error in simulations and real insurance data.

Contribution

Introduction of asymptotically unbiased, consistent, and normally distributed estimators for tail index using regularisation techniques.

Findings

Estimators exhibit low bias and MSE in simulations

Asymptotic properties confirmed analytically

Effective in insurance claim data analysis

Abstract

In this paper, we introduce reduced-bias estimators for the estimation of the tail index of a Pareto-type distribution. This is achieved through the use of a regularised weighted least squares with an exponential regression model for log-spacings of top order statistics. The asymptotic properties of the proposed estimators are investigated analytically and found to be asymptotically unbiased, consistent and normally distributed. Also, the finite sample behaviour of the estimators are studied through a simulations theory. The proposed estimators were found to yield low bias and MSE. In addition, the proposed estimators are illustrated through the estimation of the tail index of the underlying distribution of claims from the insurance industry.