$L_p$ and almost sure convergence of estimation on heavy tail index under random censoring

TL;DR

This paper establishes $L_p$ and almost sure convergence of a tail index estimator under random censoring, providing theoretical guarantees and finite sample performance insights for heavy tail analysis.

Contribution

It proves convergence properties of a tail index estimator under random censoring and quantifies its finite sample performance through simulations.

Findings

Estimator converges in $L_p$ and almost surely under certain conditions.

Error moments decay at a rate of $O(1/\log^{m\kappa/2} n)$.

Estimator performs well even with high censoring rates.

Abstract

In this paper, we prove $L_p,\ p\geq 2$ and almost sure convergence of tail index estimator mentioned in \cite{grama2008} under random censoring and several assumptions. $p$th moment of the error of the estimator is proved to be of order $O\left(\frac{1}{\log^{m\kappa/2}n}\right)$ with given assumptions. We also perform several finite sample simulations to quantify performance of this estimator. Finite sample results show that the proposed estimator is effective in finding underlying tail index even when censor rate is high.