Asymptotic normality of Hill Estimator for truncated data

TL;DR

This paper investigates the asymptotic normality of the Hill estimator for truncated data, extending known results from untruncated cases by analyzing second order regular variation.

Contribution

It establishes the asymptotic normality of the Hill estimator in the truncated data setting under second order regular variation assumptions.

Findings

Proves asymptotic normality of Hill estimator for truncated data

Extends second order regular variation results to truncated distributions

Provides theoretical foundation for tail index estimation with truncated data

Abstract

The problem of estimating the tail index from truncated data is addressed in Chakrabarty and Samorodnitsky (2009). In that paper, a sample based (and hence random) choice of k is suggested, and it is shown that the choice leads to a consistent estimator of the inverse of the tail index. In this paper, the second order behavior of the Hill estimator with that choice of k is studied, under some additional assumptions. In the untruncated situation, it is well known that asymptotic normality of the Hill estimator follows from the assumption of second order regular variation of the underlying distribution. Motivated by this, we show the same in the truncated case in light of the second order regular variation.