Nonparametric estimation of the lower tail dependence λ_L in bivariate copulas

Abstract

The lower tail dependence λ _L is a measure that characterizes the tendency of extreme co-movements in the lower tails of a bivariate distribution. It is invariant with respect to strictly increasing transformations of the marginal distribution and is therefore a function of the copula of the bivariate distribution. λ _L plays an important role in modelling aggregate financial risk with copulas. This paper introduces three non-parametric estimators for λ _L . They are weakly consistent under mild regularity conditions on the copula and under the assumption that the number k = k(n) of observations in the lower tail, used for estimation, is asymptotically k ≈ √n. The finite sample properties of the estimators are investigated using a Monte Carlo simulation in special cases. It turns out that these estimators are biased, where amount and sign of the bias depend on the underlying copula, on the sample size n, on k, and on the true value of λ _L .

Keywords:

• Copula
• lower tail dependence
• non-parametric estimation
• empirical copula process
• consistency of estimators
• small sample properties of estimators