Revisiting the Maximum Likelihood Estimation of a Positive Extreme Value Index

Abstract

In this article, we revisit Feuerverger and Hall’s maximum likelihood estimation of the extreme value index. Based on those estimators we propose new estimators that have the smallest possible asymptotic variance, equal to the asymptotic variance of the Hill estimator. The full asymptotic distributional properties of the estimators are derived under a general third-order framework for heavy tails. Applications to a real data set and to simulated data are also presented.

AMS Subject Classification:

• 62G05
• 62G20
• 62G32

Keywords:

• Bias estimation
• Heavy tails
• Semiparametric estimation
• Statistics of extremes