Nonparametric regression estimation of conditional tails: the random covariate case

Abstract

We present families of nonparametric estimators for the conditional tail index of a Pareto-type distribution in the presence of random covariates. These families are constructed from locally weighted sums of power transformations of excesses over a high threshold. The asymptotic properties of the proposed estimators are derived under some assumptions on the conditional response distribution, the weight function and the density function of the covariates. We also introduce bias-corrected versions of the estimators for the conditional tail index, and propose in this context a consistent estimator for the second-order tail parameter. The finite sample performance of some specific examples from our classes of estimators is illustrated with a small simulation experiment.

Keywords:

• tail index
• Pareto-type distribution
• regression
• kernel statistic
• bias-correction

Acknowledgements

The authors are very grateful to the two anonymous referees and the associate editor for their very constructive comments on the paper. Their suggestions have definitely improved the exposition of the material.