Estimating the second-order parameter of regular variation and bias reduction in tail index estimation under random truncation

TL;DR

This paper introduces new estimators for the second-order parameter and tail index in truncated Pareto-type distributions, demonstrating their consistency, asymptotic normality, and good finite-sample performance through simulations.

Contribution

It proposes novel estimators for the second-order parameter and tail index under random truncation, with proven asymptotic properties and improved bias reduction.

Findings

Estimators are consistent and asymptotically normal.

Simulation results show good bias and mean square error performance.

The methods extend existing extreme value theory to truncated data.

Abstract

In this paper, we propose an estimator of the second-order parameter of randomly right-truncated Pareto-type distributions data and establish its consistency and asymptotic normality. Moreover, we derive an asymptotically unbiased estimator of the tail index and study its asymptotic behaviour. Our considerations are based on a useful Gaussian approximation of the tail product-limit process recently given by Benchaira et al. [Tail product-limit process for truncated data with application to extreme value index estimation. Extremes, 2016; 19: 219-251] and the results of Gomes et al. [Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes, 2002; 5: 387-414]. We show, by simulation, that the proposed estimators behave well, in terms of bias and mean square error.