A new method for estimating the tail index using truncated sample sequence

TL;DR

This paper introduces two novel truncated estimators for accurately estimating the tail index of extremely heavy-tailed distributions, demonstrating superior performance through theoretical analysis and numerical simulations.

Contribution

The paper presents two new truncated estimators for tail index estimation and proves their asymptotic properties, improving estimation accuracy for heavy-tailed distributions.

Findings

The new estimators perform well in estimation error and power.

They outperform six existing estimators in simulations.

The estimators are effective for distributions with infinite mean or variance.

Abstract

This article proposes a new method of truncated estimation to estimate the tail index $\alpha$ of the extremely heavy-tailed distribution with infinite mean or variance. We not only present two truncated estimators $\hat{\alpha}$ and $\hat{\alpha}^{\prime}$ for estimating $\alpha$ ($0<\alpha \leq 1$) and $\alpha$ ($1<\alpha \leq 2$) respectively, but also prove their asymptotic statistical properties. The numerical simulation results comparing the six known estimators in estimating error, the Type I Error and the power of estimator show that the performance of the two new truncated estimators is quite good on the whole.