Estimating the tail index of Pareto-type distributions from geometric records

TL;DR

This paper introduces a new geometric record-based maximum likelihood estimator for the tail index of Pareto-type distributions, demonstrating superior performance in sequential and costly measurement scenarios.

Contribution

It develops a novel estimator with proven consistency and normality, extending to Pareto-type distributions and outperforming classical methods like Hill's estimator.

Findings

Estimator is strongly consistent and asymptotically normal.

Performs better than Hill's estimator with fewer measurements.

Effective in sequential and destructive testing applications.

Abstract

In this paper we develop a novel inferential approach based on geometric records for estimating the tail index of heavy-tailed distributions. We construct a maximum likelihood estimator for the Pareto model and establish its strong consistency and asymptotic normality, providing also an explicit expression for its asymptotic variance. These results are then extended to a broad class of Pareto-type distributions. The performance of the estimator is assessed via Monte Carlo simulation and compared with classical estimators from the literature. The proposed method is particularly well suited for settings where data arrive sequentially, as it yields smooth estimation trajectories. It is also especially advantageous in applications such as destructive testing, where measuring each observation exactly is costly. In this context, the estimator clearly outperforms Hill's estimator, achieving comparable or better accuracy while requiring a substantially smaller number of measured observations. An application to the analysis of the distribution of fluctuations of the Dow Jones Industrial Average (DJI) is also presented.