Local Estimation of the Second-Order Parameter in Extreme Value Statistics and Local Unbiased Estimation of the Tail Index

Abstract

We develop and study in the framework of Pareto-type distributions a class of nonparametric kernel estimators for the conditional second order tail parameter. The estimators are obtained by local estimation of the conditional second order parameter using a moving window approach. Asymptotic normality of the proposed class of kernel estimators is proven under some suitable conditions on the kernel function and the conditional tail quantile function. The nonparametric estimators for the second order parameter are subsequently used to obtain a class of bias-corrected kernel estimators for the conditional tail index. In particular it is shown how for a given kernel function one obtains a bias-corrected kernel function, and that replacing the second order parameter in the latter with a consistent estimator does not change the limiting distribution of the bias-corrected estimator for the conditional tail index. The finite sample behavior of some specific estimators is illustrated with a simulation experiment. The developed methodology is also illustrated on fire insurance claim data.

Keywords:

• Bias-correction
• Kernel estimator
• Pareto-type distribution
• Regression
• Second-order parameter

Mathematics Subject Classification:

• Primary 62G08, 62G30, 62G32
• Secondary 62E30, 62G20

Acknowledgment

The authors are very grateful to the referees for their careful reading of the article and their comments that led to significant improvements of the initial draft. The first author's research was supported by a grant from the Danish Natural Science Research Council. The second author's research was partially supported by the National Research Foundation.