Multivariate Archimedean copulas, $d$-monotone functions and $\ell_1$-norm symmetric distributions

By Alexander J. McNeil, Johanna G. Nevslehov'a. 2009

727 citations recorded by the source corpus.

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.

Subfields: Finance, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ccfd31ec9b5ae150ed56da677b68e1e800b72ef3

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Citations in the corpus

Reparameterizing Marshall–Olkin copulas with applications to sampling (2011, background): “In 6 one finds a probabilistic model for a general Archimedean copula.”
Asymptotics of empirical copula processes under non-restrictive smoothness assumptions (2010, neutral): “C(u) = φ−1(φ(u1) + · · · + φ(ud)), u ∈ [0,1]d , where the generator φ : [0,1] → [0,∞] is convex, decreasing, finite on (0,1], and vanishes at 1, whereas φ−1 : [0,∞) → [0,1] is its generalized inverse, φ−1(x) = inf{u ∈ [0,1]: φ(u) ≤ x}; in fact, if d ≥ 3, more conditions on φ are required for C to be”
Extreme-value copulas (2009, background): “Consider the Archimedean copula → { ∈ } the function f should be strictly decreasing and convex and satisfy f ( 1 ) = 0, and f ← should be d -monotone on ( 0 , ¥ ) , see [69].”
Inference in multivariate Archimedean copula models (2011, methodology): “This is a prerequisite to the construction of valid goodness-of-fit and model selection procedures for multivariate Archimedean copulas based on the representation of McNeil and Neslehova (2009) .”
Copula Theory: An Introduction (2010, methodology): “Basically, we follow the approach in [136] (see also [135]).”
ESTIMATORS BASED ON KENDALL'S TAU IN MULTIVARIATE COPULA MODELS (2011, background): “It is shown in McNeil & Nešlehová (2009) that an Archimedean copula C is the survival copula of a simplex random vector X = R V, where R is a positive random variable, independent of the d-variate random vector V which is uniformly distributed on S + 1 = {(v1, . . . , vd ) ∈ Rd+ : v1 + · · · + vd…”
Flexible dependence modeling of operational risk losses and its impact on total capital requirements (2014, background): “As vine copulas, on the other hand, do not have a closed-form copula expression, they should also be ruled out (see Panagiotelis et al. [2012] for an alternative, Pair Copula Construction (PCC)-based approach to modeling multivariate discrete data).”
Statistics and Quantitative Risk Management for Banking and Insurance (2014, neutral): “A different ansatz stems from Archimedean copulas, which are given explicitly in terms of a so-called generator ψ : [0,∞)→ [0, 1] via C(u) = ψ(ψ−1(u1) + · · ·+ ψ−1(ud)), u ∈ [0, 1]d; (8) see McNeil and Nešlehová (2009).”
Statistical inference for time-changed Lévy processes via composite characteristic function estimation (2010, methodology): “…example, one can directly adapt our method to the problem of semiparametric inference in distributional Archimedian copula-based models [see, e.g., McNeil and Nešlehová (2009) for recent results], where one faces the problem of estimating a multidimensional distribution function of the form F(x1,…”
Models for construction of multivariate dependence – a comparison study (2009, methodology): “McNeil (2007) shows how to use the Laplace-transform method also for the NACs, in the case where all generators are taken from either the Gumbel-or the Clayton family.”
Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness (2009, background): “The condition of complete monotonicity can be relaxed if one only requires that ψ generates a copula for a fixed dimension n; see McNeil-Nešlehová [18].”
CDO pricing with nested Archimedean copulas (2011, background): “McNeil and Nešlehová (2009) show that an Archimedean generator defines an exchangeable Archimedean copula, given by CðuÞ ¼ Cðu1, . . . , uI; Þ ¼ ð 1ðu1Þ þ þ 1ðuIÞÞ, u 2 ½0, 1 I, ð3Þ if and only if is I-monotone, i.e. is continuous on [0,1], has derivatives up to order I 2 satisfying ( 1)k(dk/dtk)…”