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Marius Hofert - One of the best experts on this subject based on the ideXlab platform.
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on structure family and parameter estimation of hierarchical Archimedean copulas
arXiv: Methodology, 2016Co-Authors: Jan Gorecki, Marius Hofert, Martin HoleňaAbstract:Research on structure determination and parameter estimation of hierarchical Archimedean copulas (HACs) has so far mostly focused on the case in which all appearing Archimedean copulas belong to the same Archimedean family. The present work addresses this issue and proposes a new approach for estimating HACs that involve different Archimedean families. It is based on employing goodness-of-fit test statistics directly into HAC estimation. The approach is summarized in a simple algorithm, its theoretical justification is given and its applicability is illustrated by several experiments, which include estimation of HACs involving up to five different Archimedean families.
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likelihood inference for Archimedean copulas in high dimensions under known margins
Journal of Multivariate Analysis, 2012Co-Authors: Marius Hofert, Martin Machler, Alexander J McneilAbstract:Explicit functional forms for the generator derivatives of well-known one-parameter Archimedean copulas are derived. These derivatives are essential for likelihood inference as they appear in the copula density, conditional distribution functions, and the Kendall distribution function. They are also required for several asymmetric extensions of Archimedean copulas such as Khoudraji-transformed Archimedean copulas. Availability of the generator derivatives in a form that permits fast and accurate computation makes maximum-likelihood estimation for Archimedean copulas feasible, even in large dimensions. It is shown, by large scale simulation of the performance of maximum likelihood estimators under known margins, that the root mean squared error actually decreases with both dimension and sample size at a similar rate. Confidence intervals for the parameter vector are derived under known margins. Moreover, extensions to multi-parameter Archimedean families are given. All presented methods are implemented in the R package nacopula and can thus be studied in detail.
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A stochastic representation and sampling algorithm for nested Archimedean copulas
Journal of Statistical Computation and Simulation, 2012Co-Authors: Marius HofertAbstract:A general sampling algorithm for nested Archimedean copulas was recently suggested. It is given in two different forms, a recursive or an explicit one. The explicit form allows for a simpler version of the algorithm which is numerically more stable and faster since less function evaluations are required. The algorithm can also be given in general form, not being restricted to a particular nesting such as fully nested Archimedean copulas. Further, several examples are given.
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efficiently sampling nested Archimedean copulas
Computational Statistics & Data Analysis, 2011Co-Authors: Marius HofertAbstract:Efficient sampling algorithms for both Archimedean and nested Archimedean copulas are presented. First, efficient sampling algorithms for the nested Archimedean families of Ali-Mikhail-Haq, Frank, and Joe are introduced. Second, a general strategy how to build a nested Archimedean copula from a given Archimedean generator is presented. Sampling this copula involves sampling an exponentially tilted stable distribution. A fast rejection algorithm is developed for the more general class of tilted Archimedean generators. It is proven that this algorithm reduces the complexity of the standard rejection algorithm to logarithmic complexity. As an application it is shown that the fast rejection algorithm outperforms existing algorithms for sampling exponentially tilted stable distributions involved, e.g., in nested Clayton copulas. Third, with the additional help of randomization of generator parameters, explicit sampling algorithms for several nested Archimedean copulas based on different Archimedean families are found. Additional results include approximations and some dependence properties, such as Kendall's tau and tail dependence parameters. The presented ideas may also apply in the more general context of sampling distributions given by their Laplace-Stieltjes transforms.
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sampling Archimedean copulas
Computational Statistics & Data Analysis, 2008Co-Authors: Marius HofertAbstract:The challenge of efficiently sampling exchangeable and nested Archimedean copulas is addressed. Specific focus is put on large dimensions, where methods involving generator derivatives are not applicable. Additionally, new conditions under which Archimedean copulas can be mixed to construct nested Archimedean copulas are presented. Moreover, for some Archimedean families, direct sampling algorithms are given. For other families, sampling algorithms based on numerical inversion of Laplace transforms are suggested. For this purpose, the Fixed Talbot, Gaver Stehfest, Gaver Wynn rho, and Laguerre series algorithm are compared in terms of precision and runtime. Examples are given, including both exchangeable and nested Archimedean copulas.
Johanna Neslehova - One of the best experts on this subject based on the ideXlab platform.
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inference in multivariate Archimedean copula models
Test, 2011Co-Authors: Christian Genest, Johanna Neslehova, Johanna F ZiegelAbstract:This paper proposes new rank-based estimators for multivariate Archimedean copulas. The approach stems from a recent representation of these copulas as the survival copulas of simplex distributions. The procedures are based on a reconstruction of the radial part of the simplex distribution from the Kendall distribution, which arises through the multivariate probability integral transformation of the data. In the bivariate case, the methodology is justified by the well known fact that an Archimedean copula is in one-to-one correspondence with its Kendall distribution. It is proved here that this property continues to hold in the trivariate case, and strong evidence is provided that it extends to any dimension. In addition, a criterion is derived for the convergence of sequences of multivariate Archimedean copulas. This result is then used to show consistency of the proposed estimators.
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from Archimedean to liouville copulas
Journal of Multivariate Analysis, 2010Co-Authors: Alexander J Mcneil, Johanna NeslehovaAbstract:We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Neslehova (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall's tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall's tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.
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multivariate Archimedean copulas d monotone functions and l1 norm symmetric distributions
Annals of Statistics, 2009Co-Authors: Alexander J Mcneil, Johanna NeslehovaAbstract:It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a d-dimensional copula is that the generator is a d-monotone function. The class of d-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of d-dimensional l 1 ; -norm symmetric distributions that place no point mass at the origin. The d-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189-207] in an analogous manner to the well-known Bernstein-Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean copulas. They also yield useful expressions for the d-dimensional Kendall function and Kendall's rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.
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multivariate Archimedean copulas d monotone functions and ell_1 norm symmetric distributions
arXiv: Statistics Theory, 2009Co-Authors: Alexander J Mcneil, Johanna NeslehovaAbstract:It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a $d$-dimensional copula is that the generator is a $d$-monotone function. The class of $d$-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of $d$-dimensional $\ell_1$-norm symmetric distributions that place no point mass at the origin. The $d$-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189--207] in an analogous manner to the well-known Bernstein--Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean copulas. They also yield useful expressions for the $d$-dimensional Kendall function and Kendall's rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.
Alexander J Mcneil - One of the best experts on this subject based on the ideXlab platform.
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likelihood inference for Archimedean copulas in high dimensions under known margins
Journal of Multivariate Analysis, 2012Co-Authors: Marius Hofert, Martin Machler, Alexander J McneilAbstract:Explicit functional forms for the generator derivatives of well-known one-parameter Archimedean copulas are derived. These derivatives are essential for likelihood inference as they appear in the copula density, conditional distribution functions, and the Kendall distribution function. They are also required for several asymmetric extensions of Archimedean copulas such as Khoudraji-transformed Archimedean copulas. Availability of the generator derivatives in a form that permits fast and accurate computation makes maximum-likelihood estimation for Archimedean copulas feasible, even in large dimensions. It is shown, by large scale simulation of the performance of maximum likelihood estimators under known margins, that the root mean squared error actually decreases with both dimension and sample size at a similar rate. Confidence intervals for the parameter vector are derived under known margins. Moreover, extensions to multi-parameter Archimedean families are given. All presented methods are implemented in the R package nacopula and can thus be studied in detail.
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from Archimedean to liouville copulas
Journal of Multivariate Analysis, 2010Co-Authors: Alexander J Mcneil, Johanna NeslehovaAbstract:We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Neslehova (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall's tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall's tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.
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multivariate Archimedean copulas d monotone functions and l1 norm symmetric distributions
Annals of Statistics, 2009Co-Authors: Alexander J Mcneil, Johanna NeslehovaAbstract:It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a d-dimensional copula is that the generator is a d-monotone function. The class of d-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of d-dimensional l 1 ; -norm symmetric distributions that place no point mass at the origin. The d-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189-207] in an analogous manner to the well-known Bernstein-Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean copulas. They also yield useful expressions for the d-dimensional Kendall function and Kendall's rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.
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multivariate Archimedean copulas d monotone functions and ell_1 norm symmetric distributions
arXiv: Statistics Theory, 2009Co-Authors: Alexander J Mcneil, Johanna NeslehovaAbstract:It is shown that a necessary and sufficient condition for an Archimedean copula generator to generate a $d$-dimensional copula is that the generator is a $d$-monotone function. The class of $d$-dimensional Archimedean copulas is shown to coincide with the class of survival copulas of $d$-dimensional $\ell_1$-norm symmetric distributions that place no point mass at the origin. The $d$-monotone Archimedean copula generators may be characterized using a little-known integral transform of Williamson [Duke Math. J. 23 (1956) 189--207] in an analogous manner to the well-known Bernstein--Widder characterization of completely monotone generators in terms of the Laplace transform. These insights allow the construction of new Archimedean copula families and provide a general solution to the problem of sampling multivariate Archimedean copulas. They also yield useful expressions for the $d$-dimensional Kendall function and Kendall's rank correlation coefficients and facilitate the derivation of results on the existence of densities and the description of singular components for Archimedean copulas. The existence of a sharp lower bound for Archimedean copulas with respect to the positive lower orthant dependence ordering is shown.
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sampling nested Archimedean copulas
Journal of Statistical Computation and Simulation, 2008Co-Authors: Alexander J McneilAbstract:We give algorithms for sampling from non-exchangeable Archimedean copulas created by the nesting of Archimedean copula generators, where in the most general algorithm the generators may be nested to an arbitrary depth. These algorithms are based on mixture representations of these copulas using Laplace transforms. While in principle the approach applies to all nested Archimedean copulas, in practice the approach is restricted to certain cases where we are able to sample distributions with given Laplace transforms. Precise instructions are given for the case when all generators are taken from the Gumbel parametric family or the Clayton family; the Gumbel case in particular proves very easy to simulate.
Henri Benisty - One of the best experts on this subject based on the ideXlab platform.
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GaN light-emitting diodes with Archimedean lattice photonic crystals
Applied Physics Letters, 2006Co-Authors: Aurélien David, Tetsuo Fujii, Elison Matioli, Rajat Sharma, Shuji Nakamura, Steven P. Denbaars, Claude Weisbuch, Henri BenistyAbstract:We study GaN-based light emitting diodes incorporating an omnidirectional photonic crystal with Archimedean lattice. Photonic bands are observed over several Brillouin zones, revealing reciprocal space symmetries and evidencing the omnidirectionality of the photonic crystal. Intensities of the diffracted bands are found to agree with the Fourier transform of the crystal lattice, and confirm its Archimedean nature.
Susanne Samingerplatz - One of the best experts on this subject based on the ideXlab platform.
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the dominance relation in some families of continuous Archimedean t norms and copulas
Fuzzy Sets and Systems, 2009Co-Authors: Susanne SamingerplatzAbstract:The dominance relation in several families of continuous Archimedean t-norms and copulas is investigated. On the one hand, the contribution provides a comprehensive overview on recent conditions and properties of dominance as well as known results for particular cases of families. On the other hand, it contains new results clarifying the dominance relationship in five additional families of continuous Archimedean t-norms and copulas.
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a generalization of the mulholland inequality for continuous Archimedean t norms
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Susanne Samingerplatz, Bernard De Baets, Hans De MeyerAbstract:It is well known that dominance between strict t-norms is closely related to the Mulholland inequality, which can be seen as a generalization of the Minkowski inequality. However, strict t-norms constitute only one part of the class of continuous Archimedean t-norms, the basic elements from which all continuous t-norms are composed. In this paper, dominance between continuous Archimedean t-norms is shown to be related to a generalization of the Mulholland inequality. We provide sufficient and necessary conditions for its fulfillment.
