A method of moments estimator of tail dependence

TL;DR

This paper introduces a moments-based estimator for tail dependence in multivariate extremes, providing a semi-parametric approach that is consistent, asymptotically normal, and useful for models where likelihood methods fail.

Contribution

It proposes a novel method of moments estimator for tail dependence parameters in a semi-parametric model, extending to multivariate cases and offering a goodness-of-fit test.

Findings

Estimator is consistent and asymptotically normal under weak conditions.

Performs well for models where likelihood methods are ineffective.

Provides a practical approach for multivariate tail dependence estimation.

Abstract

In the world of multivariate extremes, estimation of the dependence structure still presents a challenge and an interesting problem. A procedure for the bivariate case is presented that opens the road to a similar way of handling the problem in a truly multivariate setting. We consider a semi-parametric model in which the stable tail dependence function is parametrically modeled. Given a random sample from a bivariate distribution function, the problem is to estimate the unknown parameter. A method of moments estimator is proposed where a certain integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. Under very weak conditions, the estimator is shown to be consistent and asymptotically normal. Moreover, a comparison between the parametric and nonparametric estimators leads to a goodness-of-fit test for the semiparametric model. The performance of the estimator is illustrated for a discrete spectral measure that arises in a factor-type model and for which likelihood-based methods break down. A second example is that of a family of stable tail dependence functions of certain meta-elliptical distributions.