Intermediate Tail Dependence: A Review and Some New Results

The concept of intermediate tail dependence is useful if one wants to quantify the degree of positive dependence in the tails when there is no strong evidence of presence of the usual tail dependence. We first review existing studies on intermediate tail dependence, and then we report new results to supplement the review. Intermediate tail dependence for elliptical, extreme value and Archimedean copulas are reviewed and further studied, respectively. For Archimedean copulas, we not only consider the frailty model but also the recently studied scale mixture model; for the latter, conditions leading to upper intermediate tail dependence are presented, and it provides a useful way to simulate copulas with desirable intermediate tail dependence structures.Comment: 25 pages, 1 figur

A note on a non-parametric tail dependence estimator

We present a non-parametric tail dependence estimator which arises naturally from a specific regression model. Above that, this tail dependence estimator also results from a specific copula mixture. --Upper tail dependence,nonparametric estimation,copula

Methods of Tail Dependence Estimation

Characterization and quantification of climate extremes and their dependencies are fundamental to the studying of natural hazards. This chapter reviews various parametric and nonparametric tail dependence coefficient estimators. The tail dependence coefficient describes the dependence (degree of association) between concurrent extremes at different locations. Accurate and reliable knowledge of the spatial characteristics of extremes can help improve the existing methods of modeling the occurrence probabilities of extreme events. This chapter will review these methods and use two case studies to demonstrate the application of tail dependence analysis

A new class of copulas with tail dependence and a generalized tail dependence estimator

We present a new family of copulas (generalized mean copulas) which is positive comprehensive and allows for upper tail dependence. It includes the Spearman copula and a specific Fréchet copula as special cases. Some properties and a generalized tail dependence estimator are derived. Finally, a small simulation study is conducted. --Geometric mean,arithmetic mean,copula,tail dependence

Distorted Copulas: Constructions and Tail Dependence

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1] the distortion C ψ: [0, 1]2 → [0, 1], C ψ(x, y) = ψ{C[ψ−1(x), ψ−1(y)]} is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ that ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails

Estimating Tail Dependence of Elliptical Distributions

Recently there has been an increasing interest in applying elliptical distributions to risk management. Under weak conditions, Hult and Lindskog (2002) showed that a random vector with an elliptical distribution is in the domain of attraction of a multivariate extreme value distribution. In this paper we study two estimators for the tail dependence function, which are based on extreme value theory and the structure of an elliptical distribution, respectively. After deriving second order regular variation estimates and proving asymptotic normality for both estimators, we show that the estimator based on the structure of an elliptical distribution is better than that based on extreme value theory in terms of both asymptotic variance and optimal asymptotic mean squared error.Our theoretical results are confirmed by a simulation study

Lower Tail Dependence for Archimedean Copulas: Characterizations and Pitfalls

Tail dependence copulas provide a natural perspective from which one can study the dependence in the tail of a multivariate distribution.For Archimedean copulas with continuously differentiable generators, regular variation of the generator near the origin is known to be closely connected to convergence of the corresponding lower tail dependence copulas to the Clayton copula.In this paper, these characterizations are refined and extended to the case of generators which are not necessarily continuously differentiable.Moreover, a counterexample is constructed showing that even if the generator of a strict Archimedean copula is continuously differentiable and slowly varying at the origin, then the lower tail dependence copulas do not need to converge to the independent copula.Archimedean copula;regular variation;tail dependence;de Haan class