Asymptotically Unbiased Estimation of the Coefficient of Tail Dependence

Abstract

We introduce and study a class of weighted functional estimators for the coefficient of tail dependence in bivariate extreme value statistics. Asymptotic normality of these estimators is established under a second order condition on the joint tail behavior, some conditions on the weight function and for appropriately chosen sequences of intermediate order statistics.Asymptotically unbiased estimators are constructed by judiciously chosen linear combinations of weighted functional estimators, and variance optimality within this class of asymptotically unbiased estimators is discussed. The finite sample performance of some specific examples from our class of estimators and some alternatives from the recent literature are evaluated with a small simulation experiment.

Bias correction in multivariate extremes

“…A possible way to estimate ρ is to use on each margin one of the techniques developed in the univariate setting; see, for example, Gomes, de Haan and Peng (2002) or Ciuperca and Mercadier (2010). Other methods make use of the multivariate structure of the data; see, for example, Peng (2010) and also Goegebeur and Guillou (2013) in a slightly different framework. The construction described here takes likewise advantage of the multivariate information of the sample.…”

“…Bias correction problems in the bivariate context received less attention than in the univariate setting. To the best of our knowledge, it seems to be reduced to Beirlant, Dierckx and Guillou (2011) and Goegebeur and Guillou (2013), who consider the estimation of bivariate joint tails, which differs slightly from our task.…”

Interval estimation for a measure of tail dependence

“…Therefore, η and the limit of s(t) can be used to distinguish asymptotic dependence (i.e., η = 1& lim t→0 s(t) > 0) and asymptotic independence (i.e., η < 1 or η = 1& lim t→0 s(t) = 0). Statistical inference for η is available in Dutang et al (2014), Draisma et al (2004), Goegebeur and Guillou (2012) and Peng (1999).…”

Robust nonparametric estimation of the conditional tail dependence coefficient

“…In absence of covariates, several estimators for η have been introduced in the extreme value literature. We refer to Ledford and Tawn (1997), Peng (1999), Draisma et al (2004), Beirlant et al (2011), and Goegebeur and Guillou (2013), to name but a few.…”