a b s t r a c tA probabilistic interpretation for hierarchical Archimedean copulas based on Lévy subordinators is given. Independent exponential random variables are divided by groupspecific Lévy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas.
Associated with any parametric family of Lévy subordinators there is a parametric family of extendible Marshall-Olkin copulas, which shares the dependence structure with the vector of first passage times of the Lévy subordinator across i.i.d. exponential threshold levels. The present article derives a strongly consistent and asymptotically normal estimator for the parameters in such models. The estimation strategy is to minimize the Euclidean distance between certain empirical and theoretical functionals of the distribution. As a byproduct, the covariance structure of the order statistics of a d-dimensional extendible Marshall-Olkin distribution is computed.
A goodness-of-fit transformation for Archimedean copulas is presented from which a test can be derived. In a large-scale simulation study it is shown that the test performs well according to the error probability of the first kind and the power under several alternatives, especially in high dimensions where this test is (still) easy to apply. The test is compared to commonly applied tests for Archimedean copulas. However, these are usually numerically demanding (according to precision and runtime), especially when the dimension is large. The transformation underlying the newly proposed test was originally used for sampling random variates from Archimedean copulas. Its correctness is proven under weaker assumptions. It may be interpreted as an analogon to Rosenblatt's transformation which is linked to the conditional distribution method for sampling random variates. Furthermore, the suggested goodness-of-fit test complements a commonly used goodness-of-fit test based on the Kendall distribution function in the sense that it utilizes all other components of the transformation except the Kendall distribution function. Finally, a graphical test based on the proposed transformation is presented.
Abstract. This article presents a novel estimation procedure for high‐dimensional Archimedean copulas. In contrast to maximum likelihood estimation, the method presented here does not require derivatives of the Archimedean generator. This is computationally advantageous for high‐dimensional Archimedean copulas in which higher‐order derivatives are needed but are often difficult to obtain. Our procedure is based on a parameter‐dependent transformation of the underlying random variables to a one‐dimensional distribution where a minimum‐distance method is applied. We show strong consistency of the resulting minimum‐distance estimators to the case of known margins as well as to the case of unknown margins when pseudo‐observations are used. Moreover, we conduct a simulation comparing the performance of the proposed estimation procedure with the well‐known maximum likelihood approach according to bias and standard deviation.
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