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The lower tail dependence λL 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. λL plays an important role in modelling aggregate financial risk with copulas. This paper introduces three non-parametric estimators for λL. 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 λL. 相似文献
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Jadran Dobrić 《统计学通讯:模拟与计算》2013,42(4):1053-1068
ABSTRACT This article suggests a chi-square test of fit for parametric families of bivariate copulas. The marginal distribution functions are assumed to be unknown and are estimated by their empirical counterparts. Therefore, the standard asymptotic theory of the test is not applicable, but we derive a rule for the determination of the appropriate degrees of freedom in the asymptotic chi-square distribution. The behavior of the test under H 0 and for selected alternatives is investigated by Monte Carlo simulation. The test is applied to investigate the dependence structure of daily German asset returns. It turns out that the Gauss copula is inappropriate to describe the dependencies in the data. A t ν-copula with low degrees of freedom performs better. 相似文献
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