Matrix-tilted Archimedean copulas
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Hofert, Marius; Ziegel, Johanna F. Article Matrix-tilted Archimedean copulas Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Hofert, Marius; Ziegel, Johanna F. (2021) : Matrix-tilted Archimedean copulas, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 9, Iss. 4, pp. 1-24, https://doi.org/10.3390/risks9040068 This Version is available at: https://hdl.handle.net/10419/258157 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
risks Article Matrix-Tilted Archimedean Copulas Marius Hofert 1,* and Johanna F. Ziegel 2 Citation: Hofert, Marius, and Johanna F. Ziegel. 2021. Matrix-Tilted Archimedean Copulas. Risks 9: 68. https://doi.org/10.3390/risks9040068 Academic Editor: Mogens Steffensen Received: 5 January 2021 Accepted: 24 March 2021 Published: 6 April 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada 2Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland; [email protected] *Correspondence: [email protected] Abstract: The new class of matrix-tilted Archimedean copulas is introduced. It combines properties of Archimedean and elliptical copulas by introducing a tilting matrix in the stochastic representation of Archimedean copulas, similar to the Cholesky factor for elliptical copulas. Basic properties of this copula construction are discussed and a further extension outlined. Keywords: Archimedean copulas; elliptical copulas; stochastic representation; generalization; tilting 1. Introduction Elliptical distributions are among the most important multivariate distributions and are widely used, for example, in finance, insurance, and risk management to model multivariate risk factor changes. A d -dimensional random vector X is said to be elliptically distributed with location vector 0, if it admits the stochastic representation X=RAU, (1) where the (random) radial part R≥ 0 is independent of the random vector U which follows a uniform distribution on the Euclidean unit sphere Sk−1 and A∈Rd×k is a matrix of rank k (see Cambanis et al. (1981)); in what follows, we assume R∼FR with FR( 0 ) = 0 and k=d .Elliptical copulas are the copulas arising from elliptical distributions via Sklar’s Theorem; we assume the reader is familiar with the notion of copulas and related concepts of measuring dependence between the components of a random vector (see Nelsen 2006 for an introduction). Despite their popularity (partly due to the comparably simple simulation approach based on the stochastic representation (1) ), elliptical copulas suffer from wellknown limitations such as radial symmetry, which, in particular, implies that the lower and upper tail dependence coefficients are equal. This is often an unrealistic assumption from a practical point of view. Another popular class of copulas are Archimedean copulas, which are copulas admitting the representation C(u) = ψ(ψ−1(u1) + · · · +ψ−1(ud)),u∈[0, 1]d, for some ψ:[ 0, ∞)→[ 0, 1 ] known as (Archimedean) generator.McNeil and Nešlehová (2009) showed that a stochastic representation similar to (1) can be derived for Archimedean copulas. Archimedean copulas arise as survival copulas of random vectors X with stochastic representation X=RU, (2) where the radial part R is as in (1) , independent of U , but U is now uniformly distributed on the standard simplex ∆d={x= (x1 , . . . , xd)∈Rd|xj≥ 0, j= 1, . . . , d , x1+· · · +xd= 1 } , Risks 2021,9, 68. https://doi.org/10.3390/risks9040068 https://www.mdpi.com/journal/risks
Risks 2021,9, 68 2 of 24 that is, U∼U(∆d) . The Archimedean generator ψ corresponds to the Williamson d -transform Wd[FR]of the radial part distribution FRof Fand is given by ψ(t) = Wd[FR](t) = Z(t,∞)1−t rd−1dFR(r),t∈[0, ∞). (3) In contrast to elliptical copulas, Archimedean copulas are not restricted to radial symmetry. They are often given explicitly and properties can typically be described in terms of the generator ψ . However, one major drawback is that they are exchangeable, which implies, for example, that all pairs of random variables share the same dependence structure. This might be unrealistic from a practical point of view (see Embrechts and Hofert 2011 for a discussion). Analogously to the stochastic representation given in (1) , the main contribution of this paper is to introduce a matrix A in the stochastic representation (2) which allows one to create asymmetric copulas which are limited to neither exchangeability nor radial symmetry. The construction of this new class of copulas, referred to as matrix-tilted Archimedean copulas, is given in Section 2. Section 3discusses properties of matrix-tilted Archimedean copulas. Further extensions are mentioned in Section 4, while Section 5concludes. All proofs are provided in the Appendix A. Other novel copula constructions not further discussed here include, for example, those in (Genest et al. 2018;Krupskii and Joe 2013 2015;Quessy and Durocher 2019;Quessy et al. 2016). 2. The Construction We are interested in the survival copulas of multivariate random vectors of the form X=RAU, (4) where R and U∼U(∆d) are as above, and A= (aij)i,j=1,...,d is a (d×d) -matrix with real entries aij ∈R . The intuition behind the resulting dependence structure is the following. Naturally, our construction allows for asymmetric copulas; the degree of asymmetry could be quantified, for example, with the (scaled) supremal distance between the survival copula and copula of (4) (see Rosco and Joe 2013). For positive entries in A , mass gets squeezed towards the diagonal. For general entries, it is also possible to move mass towards the secondary diagonal. Depending on the different conditions imposed on A , the dependence structure of X can vary substantially, which can make this construction interesting for scenario generation in risk applications. For example, Figures 1and 2show samples of the empirical survival copula of X in (4) based on the same 1000 realizations of R and U but tilted with different matrices A ; note that all scatter plots of samples of such type in this paper were generated in this way. The points are colored according to the quadrant the corresponding realization of X lie in and the boundary curves of the support of the copula density is depicted.
Risks 2021,9, 68 3 of 24 ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Scatter plot of the copula of X = RAU with A = 1 0 0 1 U1 U2 ●(+,+) ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Scatter plot of the copula of X = RAU with A = 1 0.05 0.5 1 U1 U2 ●(+,+) boundary Figure 1. Scatter plots of 1000 samples from matrix-tilted Archimedean copulas with different tilting matrices A but otherwise equal realizations of R and U . The plot on the left simply corresponds to a Gumbel copula with parameter θ= 2 (Kendall’s tau 0.5). The plot on the right also displays the boundary curves (spanned by ˜e1, ˜e2; see Proposition 1below). ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Scatter plot of the copula of X = RAU with A = 1 0.05 −0.5 1 U1 U2 ● ● (+,+) (+,−) boundary ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Scatter plot of the copula of X = RAU with A = 1 −0.05 −0.5 1 U1 U2 ● ● ● (+,+) (−,+) (+,−) boundary Figure 2. Similar to Figure 1with other tilting matrices A. In what follows, we discuss a number of interesting cases and properties of matrixtilted Archimedean copulas.
Risks 2021,9, 68 4 of 24 3. Properties For the sake of tractability of the calculations and the fact that properties of multivariate copulas are often studied for bivariate margins (measures of association, tail dependence, symmetries, etc.), we focus on the bivariate case, that is, d= 2, and assume that a11 =a22 =1, a12 ≤1, a21 ≤1, a12a21 <1; (CPD) Section 3.1 addresses why Assumption (CPD) is natural. It often turns out to be useful to consider the random point ˜ U=AU, (5) so the image of U under A . All such images are denoted by their respective notation using a tilde. If A=I2 , that is, the identity matrix in the bivariate case, then ˜ U=AU=U . The columns of A are e1 , e2 in this case (the canonical basis vectors of R2 ) and our construction simply leads to the class of Archimedean copulas. In the general case, A transforms e1 , e2 to ˜e1 , ˜e2 . Figures 3and 4depict the space (randomly) spanned by (2) and (4) , corresponding to Figures 1and 2. Version March 14, 2021 submitted to Risks 4 of 24 Section 3.1 will address why Assumption (CPD) is natural. It often turns out to be useful to consider the random point ˜ U=AU,(5) so the image of U under A . All such images are denoted by their respective notation using a tilde. If 47 A=I2 , that is, the identity matrix in the bivariate case, then ˜ U=AU=U . The columns of A are 48 e1,e2 in this case (the canonical basis vectors of R2 ) and our construction simply leads to the class 49 of Archimedean copulas. In the general case, A transforms e1,e2 to ˜e1, ˜e2 . Figures 3and 4depict the 50 space (randomly) spanned by (2) and (2), corresponding to Figures 1and 2. x1 x2 0 1 1 e1 e2 x1 x2 0a11 =1 a22 =1 a12 a21 ˜e1 ˜e2 Figure 3. Depicting how (5) maps the vectors e1= (1, 0) and e2= (0, 1) to ˜e1= (1, a21 =1/2) and ˜e2= (a12 =1/20, 1), respectively, corresponding to Figure 1. x1 x2 0a11 =1 a22 =1 a12 a21 ˜e1 ˜e2 x1 x2 0a11 =1 a22 =1 a12 a21 ˜e1 ˜e2 Figure 4. Similar to Figure 3but with vectors ˜e1, ˜e2corresponding to Figure 2. 51 3.1. Special cases52 In this section, we briefly address various special cases of our construction and explain why 53 Assumption (CPD) is natural.54 First note that copulas are invariant under strictly increasing transformations of the margins. We 55 can thus always multiply each component of X by positive constants without losing flexibility of our 56 dependence model; the same scaling is commonly used for elliptical copulas where one can assume A57 to be the Cholesky factor of a correlation (instead of a more general covariance) matrix. Therefore, 58 we can equally well assume that a11 =a22 =1, the first part of Assumption (CPD). The convex cone59 between the half-lines spanned by ˜ej , j∈ {1, 2} , the columns of A , does not depend on the order of 60 Figure 3. Depicting how (5) maps the vectors e1= ( 1, 0 ) and e2= ( 0, 1 ) to ˜e1= ( 1, a21 = 1 / 2 ) and ˜e2= (a12 =1/20, 1), respectively, corresponding to Figure 1. Version March 14, 2021 submitted to Risks 4 of 24 Section 3.1 will address why Assumption (CPD) is natural. It often turns out to be useful to consider the random point ˜ U=AU,(5) so the image of U under A . All such images are denoted by their respective notation using a tilde. If 47 A=I2 , that is, the identity matrix in the bivariate case, then ˜ U=AU=U . The columns of A are 48 e1,e2 in this case (the canonical basis vectors of R2 ) and our construction simply leads to the class 49 of Archimedean copulas. In the general case, A transforms e1,e2 to ˜e1, ˜e2 . Figures 3and 4depict the 50 space (randomly) spanned by (2) and (2), corresponding to Figures 1and 2. x1 x2 0 1 1 e1 e2 x1 x2 0a11 =1 a22 =1 a12 a21 ˜e1 ˜e2 Figure 3. Depicting how (5) maps the vectors e1= (1, 0) and e2= (0, 1) to ˜e1= (1, a21 =1/2) and ˜e2= (a12 =1/20, 1), respectively, corresponding to Figure 1. x1 x2 0a11 =1 a22 =1 a12 a21 ˜e1 ˜e2 x1 x2 0a11 =1 a22 =1 a12 a21 ˜e1 ˜e2 Figure 4. Similar to Figure 3but with vectors ˜e1, ˜e2corresponding to Figure 2. 51 3.1. Special cases52 In this section, we briefly address various special cases of our construction and explain why 53 Assumption (CPD) is natural.54 First note that copulas are invariant under strictly increasing transformations of the margins. We 55 can thus always multiply each component of X by positive constants without losing flexibility of our 56 dependence model; the same scaling is commonly used for elliptical copulas where one can assume A57 to be the Cholesky factor of a correlation (instead of a more general covariance) matrix. Therefore, 58 we can equally well assume that a11 =a22 =1, the first part of Assumption (CPD). The convex cone59 between the half-lines spanned by ˜ej , j∈ {1, 2} , the columns of A , does not depend on the order of 60 Figure 4. Similar to Figure 3but with vectors ˜e1, ˜e2corresponding to Figure 2.
Risks 2021,9, 68 5 of 24 3.1. Special Cases In this section, we briefly address various special cases of our construction and explain why Assumption (CPD) is natural. First note that copulas are invariant under strictly increasing transformations of the margins. We can thus always multiply each component of X by positive constants without losing flexibility of our dependence model; the same scaling is commonly used for elliptical copulas where one can assume A to be the Cholesky factor of a correlation (instead of a more general covariance) matrix. Therefore, we can equally well assume that a11 =a22 = 1, the first part of Assumption (CPD) . The convex cone between the half-lines spanned by ˜ej , j∈ { 1, 2 } , the columns of A , does not depend on the order of the columns in A . We can thus fix the determinant to be positive, the last part of Assumption (CPD) . The assumptions a12 ≤ 1 and a21 ≤ 1 are for computational convenience. In the limiting case, when a12 =a21 = 1, we obtain det A= 0 and the corresponding matrix-tilted Archimedean copula is the upper Fréchet–Hoeffding bound M. 3.2. The Multivariate Survival Function We can now provide the survival function of X=RAUunder Assumption (CPD). Theorem 1. Under Assumption (CPD) , the joint survival function of Model (4) at (x1 , x2) is given as follows. (i) If 0≤a12 <1and 0≤a21 <1, then 1, if a21x1≥x2and a12x2≥x1, 1−a12a21 (1−a12)(1−a21)ψ(1−a21)x1+(1−a12)x2 1−a12a21 −a12 1−a12 ψx1 a12 −a21 1−a21 ψx2 a21 ,if a21x1<x2and a12x2<x1, 1 {x2≤0}+1 1−a21 ψ(x2)−a21ψx2 a21 1 {x2>0},if a21x1<x2and a12x2≥x1, 1 {x1≤0}+1 1−a12 ψ(x1)−a12ψx1 a12 1 {x1>0},if a21x1≥x2and a12x2<x1. (ii) If a12 <0and a21 <0, then 1 {x2>0}a12 1−a12 ψ(x1/a12) + 1 1−a21 ψ(x2) + 1 {x1>0}a21 1−a21 ψ(x2/a21) + 1 1−a12 ψ(x1) + 1 {x1≤0} 1 {x2≤0}a12 1−a12 ψ(x1/a12) + a21 1−a21 ψ(x2/a21) + 1,if a21x1≥x2and a12x2≥x1, 1−a12a21 (1−a12)(1−a21)ψ(1−a21)x1+(1−a12)x2 1−a12a21 ,if a21x1<x2and a12x2<x1, 1 1−a21 ψ(x2) + a12 1−a12 ψ(x1/a12),if a21x1<x2and a12x2≥x1, 1 1−a12 ψ(x1) + a21 1−a21 ψ(x2/a21),if a21x1≥x2and a12x2<x1. (iii) If 0≤a12 <1and a21 <0, then 1 {x2>0}1 1−a21 ψ(x2) + 1 {x2≤0}1+a21 1−a21 ψ(x2/a21),if a21x1≥x2and a12x2≥x1, 1−a12a21 (1−a12)(1−a21)ψ(1−a21)x1+(1−a12)x2 1−a12a21 −a12 (1−a12)ψ(x1/a12),if a21x1<x2and a12x2<x1, 1 1−a21 ψ(x2),if a21x1<x2and a12x2≥x1, a21 1−a21 ψ(x2/a21) + 1 {x1≤0} + 1 {x1>0}1 1−a12 (ψ(x1)−a12ψ(x1/a12)),if a21x1≥x2and a12x2<x1.
Risks 2021,9, 68 6 of 24 (iv) If a12 =1and 0≤a21 <1, then 1, if a21x1≥x2≥x1, 1 1−a21 1−x2 x1¯ FR(x1) + x2 x1ψ(x1)−a21 1−a21 ψ(x2/a21),if a21x1<x2<x1, 1 1−a21 (ψ(x2)−a21ψ(x2/a21)),if a21x1<x2and x2≥x1, ¯ FR(max{0, x1}),if a21x1≥x2and x2<x1. (v) If a12 =1and a21 <0, then 1 {x2≤0}1+a21 1−a21 ψ(x2/a21)+ 1 {x2>0}1 1−a21 ψ(x2),if a21x1≥x2≥x1, 1 1−a21 1−x2 x1¯ FR(x1) + x2 x1ψ(x1),if a21x1<x2<x1, 1 1−a21 ψ(x2),if a21x1<x2and x2≥x1, ¯ FR(max{0, x1}) + a21 1−a21 ψ(x2/a21),if a21x1≥x2and x2<x1. The remaining cases can be obtained by exchanging the roles of a12,a21 and x1,x2. The following corollary considers two interesting special cases for the parameters a12 and a21. Corollary 1. Suppose Assumption (CPD) holds. Then, the joint survival function of Model (4) is given as follows. (i) If a21 =0and a12 <1, then 1, if 0≥x2and a12x2≥x1, 1 1−a12 ψ(x1+ (1−a12)x2)−a12 (1−a12)ψ(x1/a12),if 0<x2and a12x2<x1, 1 1−a21 ψ(x2),if 0<x2and a12x2≥x1, 1 {x1≤0}+ 1 {x1>0}1 1−a12 (ψ(x1)−a12ψ(x1/a12)),if 0≥x2and a12x2<x1. (ii) If a12 =a21 =a, then, for 0≤a<1, we have 1, if ax1≥x2and ax2≥x1, 1 1−a(1+a)ψx1+x2 1+a−aψx1 a−aψx2 a,if ax1<x2and ax2<x1, 1 {x2≤0}+1 1−aψ(x2)−aψx2 a 1 {x2>0},if ax1<x2and ax2≥x1, 1 {x1≤0}+1 1−aψ(x1)−aψx1 a 1 {x1>0},if ax1≥x2and ax2<x1, and, for a ∈(−1, 0), we obtain that 1 {x2>0}a 1−aψ(x1/a) + 1 1−aψ(x2) + 1 {x1>0}a 1−aψ(x2/a) + 1 1−aψ(x1) + 1 {x1≤0} 1 {x2≤0}a 1−aψ(x1/a) + a 1−aψ(x2/a) + 1,if ax1≥x2and ax2≥x1, 1+a 1−aψx1+x2 1+a,if ax1<x2and ax2<x1, 1 1−aψ(x2) + a 1−aψ(x1/a),if ax1<x2and ax2≥x1, 1 1−aψ(x1) + a 1−aψ(x2/a),if ax1≥x2and ax2<x1. 3.3. The Boundary of the Support of the Copula Density The following proposition provides almost sure boundaries for X2 , the second component of X in (4) . The statements are directly clear from Figures 3and 4. The boundaries
Risks 2021,9, 68 7 of 24 carry over to the boundaries for U in the unit square after the map with the correct marginal survival functions (see also Figures 1and 2). Proposition 1. Under Assumption (CPD) , the following inequalities hold almost surely for the component X2of X= (X1,X2)of Model (4). (1) If a12 <0, X2≥x/a12 for all x <0and X2≥a21x for all x ≥0. (2) If a12 =0, X2≥a21x for all x ≥0. (3) If 0<a12 ≤1, a21x≤X2≤x/a12 for all x ≥0. 3.4. Tail Dependence In this section, we address the notion of tail dependence for the multivariate model as given in (4). Tail dependence is a copula property, so the coefficients of tail dependence only depend on the copula of (4). For the following result, note that a function h:( 0, ∞)→R is called regularly varying at x0∈[0, ∞]of index α∈R(denoted by h∈RVα) if lim x→x0 h(tx) h(x)=tα,t>0. Proposition 2. Let Assumption (CPD) hold and suppose that a12 =a21 =a . Furthermore, let the marginal distribution of X1and X2be denoted by F =F1=F2. (1) Let a ∈(0, 1). If ¯ F−(0)<∞, then λL=0. If ¯ F−(0) = ∞and ψ0∈RV−αat ∞, then λL=22−α(1+a)α−aα 1−aα. If ψ0∈RV−αat 0, then λU=21−2−α(1+a)α 1−aα. (2) Let a ∈(−1, 0). If ¯ F−(0)<∞, then λL=0. If ¯ F−(0) = ∞and ψ0∈RV−αat ∞, then λL=21+a 2α. Furthermore, λU=0. Figure 5shows λL and λU of Part (1) of Proposition 2as functions of the parameter a for various α. Assessing λL and λU in the case F16=F2 is more difficult. At least, the symmetric case a12 =a21 can provide bounds. For example, if ¯ F− 1(u)≤¯ F− 2(u) for all u sufficiently small, then (compare with the proof of Proposition 2) λL=lim u↓0 C(u,u) u=lim u↓0 P(X1>¯ F− 1(u),X2>¯ F− 2(u)) P(X1>¯ F− 1(u)) ≤lim u↓0 P(X1>¯ F− 1(u),X2>¯ F− 1(u)) P(X1>¯ F− 1(u)) (6) and one can proceed as in the proof of Proposition 2to evaluate (6) ; if ¯ F− 1(u)≥¯ F− 2(u) for all usufficiently small, then the inequality leads to a lower bound for λL.
Risks 2021,9, 68 8 of 24 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a λL α = 1 α = 4 α = 8 α = 12 α = 16 α = 20 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 a λU α = 1 α = 0.8 α = 0.6 α = 0.4 α = 0.2 α = 0.01 Figure 5. λLand λUof Part (1) of Proposition 2as functions of the parameter afor various α. 4. Randomization and Other Extensions As shown above, tilting an Archimedean copula through a matrix A makes the model more flexible by introducing different kinds of asymmetry. It is also possible to modify tail dependence properties through A . However, some parameter choices reduce the support of the copula to strict subsets of the unit square. This might be a desired feature for certain applications (a classical example is the copula of (X1 , X2) with X1≤X2 almost surely), but equally well unnatural in others. A possible remedy may be to randomize the tilting matrix A ; theoretical properties can then be derived by conditioning on A . While such random-matrix-tilted Archimedean copulas allow for very flexible models, inference is an open question. As a first example, we revisit the right-hand side of Figure 1and randomize the entry a21 = 0.5 of A with a standard uniform distribution. The result is shown on the left-hand side of Figure 6. Note that, although hardly visible here, the lower boundary as depicted on the right-hand side of Figure 1still exists but the upper one is “washed out”. If we randomize both a12 and a21 , then both boundaries disappear. The right-hand side of Figure 6 shows such a case, where a12 and a21 are (independently) randomized by Beta( 1 / 10, 1 ) and Beta( 2, 1 ) distributions, respectively. Note that both of these randomizations satisfy Assumption (CPD). As another example, let us consider the right-hand side of Figure 2and randomize the entry a21 =− 0.05 of A via a uniform distribution on [− 10, 0 ] (see the left-hand side of Figure 7) or via a21 =− 0.1 −min{E , 20 } with E being standard exponential (see the right-hand side of Figure 7). As above, both randomizations satisfy Assumption (CPD). Another option not discussed here would be to make the randomized a12 , a21 dependent by another copula. A different approach to tilting (2) by a matrix A that we do not explore is the following. Consider a random vector X=RAU, (7) where R is a radial part as in (1) , U is uniformly distributed on the `1 -sphere {x= (x1 , . . . , xd)∈Rd| |x1|+· · · +|xd|= 1 } independently of R , and A is a d×d -matrix. Then, one could consider the copula of the componentwise modulus |X| of X . Figures 8and 9 show the copulas of |X| for R and A as chosen in Figures 2and 7, respectively, and X given by (7) . One observes that, for a deterministic tilting matrix A (see Figure 8), the construction does not seem to yield anything visually much different from a Gumbel copula (in distribution). We do not have proof of this fact but we have seen this behavior in all simulation examples.
Risks 2021,9, 68 15 of 24 By distinguishing the cases a21 <0, a21 =0, and a21 >0, we obtain that E=Z(0,∞) 1 nr≤x2 a21 o 1 {r>max{x1,x2}} dFR(r) 1 {x2<0} 1 {a21<0} +Z(0,∞) 1 {r>x1}dFR(r) 1 {x2≤0} 1 {a21=0} +Z(0,∞) 1 nr>x2 a21 o 1 {r>x1}dFR(r) 1 {a21>0}. Note that, if x2≤ 0, then 1 {r>max{x1,x2}} = 1 {r>max{0,x1,x2}} = 1 {r>max{0,x1}} . This implies that E=Z(0,∞) 1 nmax{0,x1}<r≤x2 a21 odFR(r) 1 {x2<0} 1 {a21<0} +¯ FR(max{0, x1}) 1 {x2≤0} 1 {a21=0}+¯ FRmaxn0, x1,x2 a21 o 1 {a21>0} =¯ FR(max{0, x1})−¯ FRx2 a21 1 nx2 a21 >max{0,x1}o 1 {x2<0} 1 {a21<0} +¯ FR(max{0, x1}) 1 {x2≤0} 1 {a21=0}+¯ FRmaxn0, x1,x2 a21 o 1 {a21>0}. By splitting up Fin a similar fashion, we obtain that F=Z(0,∞) 1 nr>maxn0,x1,x2,x2 a21 oo1−x2 rdFR(r) 1 {a21<0} +Z(0,∞) 1 {r>max{x1,x2}}1−x2 rdFR(r) 1 {x2>0} 1 {a21=0} +Z(0,∞) 1 nmax{x1,x2}<r≤x2 a21 o1−x2 rdFR(r) 1 {x2>0} 1 {a21>0}. Distinguishing the cases of different sign of x2in the last integral leads to F=Z(x2,∞) 1 {r>x1}1−x2 rdFR(r) 1 {x2>0} 1 {a21<0} +Z(x2/a21,∞) 1 {r>x1}1−x2 rdFR(r) 1 {x2≤0} 1 {a21<0} +Z(x2,∞) 1 {r>x1}1−x2 rdFR(r) 1 {x2>0} 1 {a21=0} +Z(x2,∞) 1 nmax{0,x1}<r≤x2 a21 o1−x2 rdFR(r) 1 {x2>0} 1 {a21>0} · 1 nmax{0,x1}<x2 a21 o. Note that the first and third integrald on the right-hand side of the last equation can be pulled together. In addition, note that the second integral can be written as a21Z(x2/a21,∞)1 a21 −1 1 {r>x1}dFR(r) +Z(x2/a21,∞) 1 {r>x1}1−x2/a21 rdFR(r) 1 {x2≤0} 1 {a21<0}. Applying Lemma A1 leads to F=1−min{x1,x2} x1¯ FR(max{0, x1}) + minnx2 max{0, x1}, 1o
Risks 2021,9, 68 16 of 24 ·ψ(max{x1,x2}) 1 {x2>0} 1 {a21≤0} +1−min{max{0, x1},x2} max{0, x1}¯ FR(max{0, x1}) +minnx2 max{0, x1}, 1oψ(max{0, x1,x2}) −1−min{x2,x2/a21} x2/a21 ¯ FR(max{0, x2/a21}) −minnx2 max{0, x2/a21}, 1oψmaxnx2,x2 a21 o 1 {x2>0} 1 {a21>0} · 1 nmax{0,x1}<x2 a21 o +a211 a21 −1¯ FRmaxnx1,x2 a21 o+1−min{x1,x2/a21} x1 ·¯ FR(max{0, x1}) + minnx2/a21 max{0, x1}, 1oψmaxnx1,x2 a21 o · 1 {x2≤0} 1 {a21<0} =1−min{x1,x2} x1¯ FR(max{0, x1}) + minnx2 max{0, x1}, 1o ·ψ(max{x1,x2}) 1 {x2>0} −(1−a21)¯ FR(x2/a21) + a21ψx2 a21 1 {x2>0} 1 {a21>0} 1 nx1<x2 a21 o +a211 a21 −1¯ FRmaxnx1,x2 a21 o+1−min{x1,x2/a21} x1 ·¯ FR(max{0, x1}) + minnx2/a21 max{0, x1}, 1oψmaxnx1,x2 a21 o · 1 {x2≤0} 1 {a21<0}. Recalling that the result is E+F/(1−a21), we obtain the form as claimed. Case 2: a12 <1 and a21 =1. This case directly follows from Case 1 by interchanging a12 and a21 , as well as x1 and x2. Case 3: a12 <1 and a21 <1. Since P(RU1+a12RU2>x1 , a21RU1+RU2>x2) = R(0,∞)P(U1+a12U2>x1/r , a21U1+U2>x2/r)dFR(r) , let us first consider the integrand. By Proposition A1, we have that P(U1+a12U2>x1/r,a21U1+U2>x2/r) = A−B−C, where A=1−a12a21 −(1−a21)x1 r−(1−a12)x2 r (1−a12)(1−a21) 1 {x1<r} 1 {x2<r} · 1 nx1(1−a21)+x2(1−a12) 1−a12a21 <ro, B=a12 −x1 r 1−a12 1 {x1≤ra12} 1 {x2<r},C=a21 −x2 r 1−a21 1 {x1<r} 1 {x2≤ra21}.
Risks 2021,9, 68 17 of 24 It is straightforward to check that a21x1≥x2⇐⇒ x1(1−a21) + x2(1−a12) 1−a12a21 ≤x1, (A1) We now consider four different cases. Case 3.1: a21x1≥x2and a12x2≥x1. Condition (A1) and the last equivalence imply that 1 {x1<r} 1 nx1(1−a21)+x2(1−a12) 1−a12a21 <ro= 1 {x1<r}, so that A=1−a12a21 −(1−a21)x1 r−(1−a12)x2 r (1−a12)(1−a21) 1 {x1<r} 1 {x2<r}. This can be rewritten as A=1+1 1−a12 a12 −x1 r+1 1−a21 a21 −x2 r 1 {x1<r} 1 {x2<r} = 1 {x1<r} 1 {x2<r} +1 1−a12 a12 −x1 r 1 {x1<r} 1 {x2<r} 1 {a12≥0} +1 1−a12 a12 −x1 r 1 {x1<r} 1 {x2<r} 1 {a12<0} +1 1−a21 a21 −x2 r 1 {x1<r} 1 {x2<r} 1 {a21≥0} +1 1−a21 a21 −x2 r 1 {x1<r} 1 {x2<r} 1 {a21<0}. Furthermore, x1≤a12x2implies that 1 {a12≥0} 1 {x2<r} 1 {x1<ra12}= 1 {a12≥0} 1 {x2<r}, so that B=1 1−a12 a12 −x1 r 1 {a12≥0} 1 {x1≤ra12}+ 1 {a12<0} 1 {x1≤ra12} 1 {x2<r} =1 1−a12 a12 −x1 r 1 {a12≥0} 1 {x2<r}+ 1 {a12<0} 1 nx2<r≤x1 a12 o. Similarly, we obtain that C=1 1−a21 a21 −x2 r 1 {a21≥0} 1 {x1<r}+ 1 {a21<0} 1 nx1<r≤x2 a21 o. Summarizing the terms leads to A−B−C= 1 {x1<r} 1 {x2<r} + 1 {a12≥0} 1 1−a12 a12 −x1 r 1 {x1<r} 1 {x2<r}− 1 {x2<r} + 1 {a12<0} 1 1−a12 a12 −x1 r 1 {x1<r} 1 {x2<r}− 1 nx2<r≤x1 a12 o + 1 {a21≥0} 1 1−a21 a21 −x2 r 1 {x1<r} 1 {x2<r}− 1 {x1<r} + 1 {a21<0} 1 1−a21 a21 −x2 r 1 {x1<r} 1 {x2<r}− 1 nx1<r≤x2 a21 o.
Risks 2021,9, 68 18 of 24 Since 1 nx2<r≤x1 a12 o= 1 {x2<r} 1 nr≤x1 a12 oand 1 nx1<r≤x2 a21 o= 1 {x1<r} 1 nr≤x2 a21 o, we have A−B−C= 1 {x1<r} 1 {x2<r} +1 1−a12 a12 −x1 r 1 {x2<r} 1 {a12<0}( 1 {x1<r} − 1 nr≤x1 a12 o)− 1 {a12≥0} 1 {x1≥r} +1 1−a21 a21 −x2 r 1 {x1<r} 1 {a21<0}( 1 {x2<r} − 1 nr≤x2 a21 o)− 1 {a21≥0} 1 {x2≥r}. Consider the case a12 ≥ 0. If a12 = 0, then x1≤a12x2= 0, so that 1 {x1≥r}= 0. If a12 > 0 and x1> 0, then our above assumptions imply that x1>a12a21x1≥a12x2≥x1 , a contradiction. Thus, x1≤0 if a12 ≥0. Similarly, x2≤0 if a21 ≥0. This implies that A−B−C= 1 {x1<r} 1 {x2<r} +1 1−a12 a12 −x1 r 1 {a12<0} 1 {x1<r}− 1 nr≤x1 a12 o 1 {x2<r} +1 1−a21 a21 −x2 r 1 {a21<0} 1 {x2<r}− 1 nr≤x2 a21 o 1 {x1<r}. For a12 <0, we have x1≤a12x2≤0 if x2≥0. Thus, for a12 <0, we have that 1 {x1<r}− 1 nr≤x1 a12 o 1 {x2<r}= 1 {x1<r}− 1 nr≤x1 a12 o 1 {x2<r} 1 {x1>0} 1 {x2≥0} + 1 {x1<r}− 1 nr≤x1 a12 o 1 {x2<r} 1 {x1≤0} 1 {x2≥0} + 1 {x1<r}− 1 nr≤x1 a12 o 1 {x2<r} 1 {x1>0} 1 {x2<0} + 1 {x1<r}− 1 nr≤x1 a12 o 1 {x2<r} 1 {x1≤0} 1 {x2<0} =0+1− 1 nr≤x1 a12 o·1·1· 1 {x2≥0} + ( 1 {x1<r}−0)·1· 1 {x1>0} 1 {x2<0} +1− 1 nr≤x1 a12 o·1· 1 {x1≤0} 1 {x2<0} = 1 nr>x1 a12 o 1 {x2≥0}+ 1 {x1<r} 1 {x1>0} 1 {x2<0} + 1 nr>x1 a12 o 1 {x1≤0} 1 {x2<0}. Similarly, 1 {x2<r}− 1 nr≤x2 a21 o 1 {x1<r} = 1 nr>x2 a21 o 1 {x1≥0}+ 1 {x2<r} 1 {x1<0} 1 {x2>0}+ 1 nr>x2 a21 o 1 {x1<0} 1 {x2≤0}. Integrating the terms now leads to Z(0,∞)(A−B−C)dFR(r) =Z(0,∞) 1 {r>max{x1,x2,0}} dFR(r) + 1 {a12<0} 1 {x2≥0} 1 1−a12 Z(0,∞) 1 nr>x1 a12 oa12 −x1 rdFR(r)
Risks 2021,9, 68 19 of 24 + 1 {a12<0} 1 {x1>0} 1 {x2<0} 1 1−a12 Z(0,∞) 1 {r>x1}a12 −x1 rdFR(r) + 1 {a12<0} 1 {x1≤0} 1 {x2<0} 1 1−a12 Z(0,∞) 1 nr>x1 a12 oa12 −x1 rdFR(r) + 1 {a21<0} 1 {x1≥0} 1 1−a21 Z(0,∞) 1 nr>x2 a21 oa21 −x2 rdFR(r) + 1 {a21<0} 1 {x1<0} 1 {x2>0} 1 1−a21 Z(0,∞) 1 {r>x2}a21 −x2 rdFR(r) + 1 {a21<0} 1 {x1<0} 1 {x2≤0} 1 1−a21 Z(0,∞) 1 nr>x2 a21 oa21 −x2 rdFR(r) =¯ FR(max{x1,x2, 0}) + 1 {a12<0} 1 {x2≥0} a12 1−a12 ψ(x1/a12) + 1 {a12<0} 1 {x1>0} 1 {x2<0}1 1−a12 ψ(x1)−¯ FR(max{x1, 0}) + 1 {a12<0} 1 {x1≤0} 1 {x2<0} a12 1−a12 ψ(x1/a12) + 1 {a21<0} 1 {x1≥0} a21 1−a21 ψ(x2/a21) + 1 {a21<0} 1 {x1<0} 1 {x2>0}1 1−a21 ψ(x2)−¯ FR(max{x2, 0}) + 1 {a21<0} 1 {x1<0} 1 {x2≤0} a21 1−a21 ψ(x2/a21). Case 3.2: a21x1<x2and a12x2<x1. It is straightforward to check that, in this case, x1(1−a21) + x2(1−a12) 1−a12a21 >max{x1,x2} so that A=1−a12a21 −(1−a21)x1 r−(1−a12)x2 r (1−a12)(1−a21) 1 nx1(1−a21)+x2(1−a12) 1−a12a21 <ro =1−a12a21 (1−a12)(1−a21)1−(1−a21)x1+ (1−a12)x2 r(1−a12a21) 1 nx1(1−a21)+x2(1−a12) 1−a12a21 <ro. For B , note that a12 < 0 and x1≤ra12 imply that x2>x1 a12 ≥r . For a12 = 0, x1>a12x2=0. Furthermore, a12 >0, and x1≤ra12 imply that x2<x1 a12 ≤r. Therefore, 1 {x1≤ra12} 1 {x2<r} 1 {a12<0}=0, 1 {x1≤ra12} 1 {x2<r} 1 {a12=0}= 1 {x1≤0} 1 {x2<r} 1 {a12=0}=0, 1 {x1≤ra12} 1 {x2<r} 1 {a12>0}= 1 {x1≤ra12} 1 {a12>0}. Thus, B=a12 −x1 r 1−a12 1 {x1≤ra12} 1 {x2<r}=a12 −x1 r 1−a12 ( 1 {x1≤ra12} 1 {x2<r} 1 {a12<0} + 1 {x1≤ra12} 1 {x2<r} 1 {a12=0}+ 1 {x1≤ra12} 1 {x2<r} 1 {a12>0}) =a12 −x1 r 1−a12 1 nx1 a12 ≤ro 1 {a12>0}.
Risks 2021,9, 68 20 of 24 Similarly, C=a21 −x2 r 1−a21 1 nx2 a21 ≤ro 1 {a21>0}. Note that, in our case here, (1−a21)x1+ (1−a12)x2≥0 because 0>(1−a21)x1+ (1−a12)x2⇐⇒ x1+x2<x1a21 +x2a12, but the right-hand side is less than or equal to x1+x2 . In addition, if a12 > 0, we have that x1≥ 0, because, otherwise ( x1< 0), we have x2<x1 a12 < 0, which then leads to (1−a21)x1+ (1−a12)x2<0, a contradiction (note that a12a21 <1). This implies that Z(0,∞)(A−B−C)dFR(r) = 1−a12a21 (1−a12)(1−a21)ψ(1−a21)x1+ (1−a12)x2 1−a12a21 −a12 1−a12 1 {a12>0} 1 {x1≥0}ψx1 a12 −a21 1−a21 1 {a21>0} 1 {x2≥0}ψx2 a21 . Case 3.3: a21x1<x2and a12x2≥x1. Let x2<r . If a12 ≥ 0, then x1≤a12x2<a12r≤r , so 1 {x1<r}= 1. If a12 < 0, then a21x1<x2≤x1/a12 , thus (a21 − 1 /a12)x1≤ 0 and therefore (a12a21 − 1 )x1≥ 0. This implies that x1≤ 0 and thus that 1 {x1<r}= 1, so this indicator drops out of the term A . In addition, note that a12x2≥x1 implies x1( 1 −a21) + x2( 1 −a12)≤x2( 1 −a12a21)< r( 1 −a12a21) for x2<r . Hence, the corresponding indicator in the term A is also always one. We therefore obtain that A=1−a12a21 −(1−a21)x1 r−(1−a12)x2 r (1−a12)(1−a21) 1 {x2<r}. By considering the cases of different signs for a12, we can write the term Bas B=1 1−a12 a12 −x1 r 1 {x1≤ra12} 1 {x2<r} =1 1−a12 a12 −x1 r 1 {a12≥0} 1 {x1≤ra12} 1 {x2<r}+ 1 {a12<0} 1 {x1≤ra12} 1 {x2<r}. If x2<r and a12 ≥ 0, then x1≤a12x2<a12r , so 1 {a12≥0} 1 {x1≤ra12} 1 {x2<r}= 1 {a12≥0} 1 {x2<r}= (1− 1 {a12<0}) 1 {x2<r}. By summarizing the indicators, this yields B=1 1−a12 a12 −x1 r 1 {x2<r}− 1 {a12<0} 1 {x2<r} 1 {x1>ra12}. Since x2≤x1/a12 <rif a12 <0 and ra12 <x1, it follows that B=1 1−a12 a12 −x1 r 1 {x2<r}− 1 {a12<0} 1 {x1/a12<r}. Similarly, for C, C=1 1−a21 a21 −x2 r 1 {a21>0} 1 {x1<r} 1 {x2≤ra21}+ 1 {a21≤0} 1 {x1<r} 1 {x2≤ra21}. Note that the indicator function 1 {x1<r} drops out, since, if a21 > 0 and x2≤ra21 , then x1<x2/a21 ≤r . The second summand of indicators completely vanishes, since, if
Risks 2021,9, 68 21 of 24 a21 < 0 and x2≤ra21 , then a21x1<x2≤ra21 implies x1>r . If a21 = 0, the contradiction 0<x2≤0 follows. We thus obtain that C=1 1−a21 a21 −x2 r 1 {a21>0} 1 {x2≤ra21}. Pulling the terms together, we obtain that A−B−C =1 1−a21 1−x2 r 1 {x2<r}+1 1−a12 a12 −x1 r 1 {a12<0} 1 {x1/a12<r} −1 1−a21 a21 −x2 r 1 {a21>0} 1 {x2≤ra21}. Distinguishing the cases x2≤0 and x2>0, we obtain that A−B−C =1 1−a21 1−x2 r 1 {x2≤0} 1 {x2<r}−a21 −x2 r 1 {x2≤0} 1 {a21>0} 1 {x2≤ra21} + 1 {x2>0}1−x2 r 1 {x2<r}−a21 −x2 r 1 {a21>0} 1 {x2≤ra21} +1 1−a12 a12 −x1 r 1 {a12<0} 1 nx1 a12 <ro. Note that 1 {x2≤0} 1 {x2<r}= 1 {x2≤0} and 1 {x2≤0} 1 {a21>0} 1 {x2≤ra21}= 1 {x2≤0} 1 {a21>0} = 1 {x2≤0}( 1 − 1 {a21≤0}) . We show next that 1 {x2≤0} 1 {a21≤0}= 0. To this end, let x2≤ 0. If a21 < 0, then a21x1<x2≤ 0 implies that x1> 0 and thus a12x2≥x1> 0. If x2= 0, then a contradiction follows immediately. Otherwise, we see that a12 < 0. This yields a12a21 > 1 in contradiction to the assumption that 1 −a12a21 > 0. If a21 = 0, then 0 =a21x1<x2≤ 0 yields a contradiction immediately. Therefore, 1 {x2≤0} 1 {a21≤0}= 0 and hence 1 {x2≤0} 1 {a21>0} 1 {x2≤ra21}= 1 {x2≤0} . Similarly, if a12 < 0 and x1/a12 <r , we show that x1< 0. To this end, let x1≥ 0. Then, a12x2≥x1≥ 0, so that x2≤ 0, and thus a21x1<x2≤ 0. The case x1= 0 directly leads to a contradiction. If x1> 0, then a21 < 0, which implies that a12x2≥x1>x2/a21 . As above, this leads to a contradiction to the assumption that 1 −a12a21 > 0. Thus, 1 {a12<0} 1 nx1 a12 <ro= 1 {a12<0} 1 nx1 a12 <ro 1 {x1<0} . Plugging in these relations and summarizing the terms, we obtain that A−B−C = 1 {x2≤0}+ 1 {x2>0} 1−a21 1−x2 r 1 {r>x2}−a21 −x2 r 1 {a21>0} 1 {x2≤ra21} +1 1−a12 a12 −x1 r 1 {a12<0} 1 nx1 a12 <ro 1 {x1<0}. Integrating the terms now leads to Z(0,∞)(A−B−C)dFR(r) = 1 {x2≤0}+1 1−a21 ψ(x2)−a21 1−a21 ψx2 a21 1 {a21>0} 1 {x2>0} +a12 1−a12 ψx1 a12 1 {a12<0} 1 {x1<0}. Case 3.4: a21x1≥x2and a12x2<x1.
Risks 2021,9, 68 22 of 24 Interchanging the roles of a21,a12 and x1,x2, we obtain that Z(0,∞)(A−B−C)dFR(r) = 1 {x1≤0}+1 1−a12 ψ(x1)−a12 1−a12 ψx1 a12 1 {a12>0} 1 {x1>0} +a21 1−a21 ψx2 a21 1 {a21<0} 1 {x2<0}. Finally, here is the proof of Proposition 2. Proof of Proposition 2.(1) Recall that Ud = ( ¯ F1(X1) , ¯ F2(X2)) for X= (X1 , X2) as in (4) . Then, C(u,u) u=P(¯ F1(X1)≤u,¯ F2(X2)≤u) P(¯ F1(X1)≤u)=P(X1>¯ F− 1(u),X2>¯ F− 2(u)) P(X1>¯ F− 1(u)) . Under the given assumptions, note that the distribution of (X1 , X2) is supported in the first quadrant, the marginal survival functions are equal (denoted by ¯ F ), and, for x≥0, we have by Corollary 1that P(X1>x,X2>x) = 2 1−a1+a 2ψ2 1+ax−aψx a, (A2) ¯ F(x) = P(X1>x) = P(X2>x) = 1 1−aψ(x)−aψx a. (A3) Therefore, we obtain that λL=lim u↓0 C(u,u) u=2 lim x↑¯ F−(0) 1+a 2ψ2 1+ax−aψx a ψ(x)−aψx a. (A4) Let ¯ F−(0)<∞. Since ( 1 +a)/ 2 ∈( 1 / 2, 1 ) and ψ is non-increasing, 1+a 2ψ(2 1+ax)<ψ(x) for all x∈[ 0, ¯ F−( 0 )) . Furthermore, there exists an ˜ x∈[ 0, ¯ F−( 0 )) such that the numerator of (A4) is zero but the denominator is greater than zero for all x∈[˜ x , ¯ F−( 0 )) . Hence λL=0. If ¯ F−(0) = ∞, applying l’Hôpital’s Rule leads to λL=2 lim x↑∞ ψ02x 1+a−ψ0x a ψ0(x)−ψ0x a. Dividing by ψ0(x)and using regular variation, we obtain that λL=2 lim x↑∞ ψ0(x·2/(1+a)) ψ0(x)−ψ0(x/a) ψ0(x) 1−ψ0(x/a) ψ0(x) =21+a 2α−aα 1−aα.
Risks 2021,9, 68 23 of 24 Now, consider λUand note that λU=lim u↑1 1−2u+C(u,u) 1−u=2−lim u↑1 1−C(u,u) 1−u =2−lim u↑1 1−P(X1>¯ F−(u),X2>¯ F−(u)) 1−P(X1>¯ F−(u)) =2−lim x↓0 1−P(X1>x,X2>x) 1−P(X1>x). (A5) Using (A2) and (A3) , and proceeding similarly as above (with regular variation at 0), one obtains λUas stated. (2) Under the given assumptions, using Corollary 1, we obtain that P(X1>x,X2>x) = (2a 1−aψ(x/a) + 1, if x≤0, 1+a 1−aψ(2x/(1+a)), if x>0, ¯ F(x) = P(X1>x) = P(X2>x) = (a 1−aψ(x/a) + 1, if x≤0, 1 1−aψ(x), if x>0. This yields λL=lim u↓0 C(u,u) u=lim x↑¯ F−(0) (1+a)ψ(2x/(1+a)) ψ(x). If ¯ F−( 0 )<∞ , then λL= 0. If ¯ F−( 0 ) = ∞ , then l’Hôpital’s Rule and regular variation imply that λL=2 lim x↑∞ ψ02 1+ax ψ0(x)=22 1+a−α. Now, consider λU and note that the margins live on R . Similar to above, we obtain that λU=2−lim x↓−∞ 1−P(X1>x,X2>x) 1−P(X1>x) which is easily seen to be 0. References Cambanis, Stamatis, Steel Huang, and Gordon Simons. 1981. On the theory of elliptically contoured distributions. Journal of Multivariate Analysis 11: 368–85. [CrossRef] Embrechts, Paul, and Marius Hofert. 2011. Comments on: Inference in multivariate Archimedean copula models. TEST 20: 263–70. [CrossRef] Genest, Christian, Johanna Nešlehová, and Louis-Paul Rivest. 2018. The class of multivariate max-id copulas with `1 -norm symmetric exponent measure. Bernoulli 24: 3751–90. [CrossRef] Krupskii, Pavel, and Harry Joe. 2013. Factor copula models for multivariate data. Journal of Multivariate Analysis 120: 85–101. [CrossRef] Krupskii, Pavel, and Harry Joe. 2015. Structured factor copula models: Theory, inference and computation. Journal of Multivariate Analysis 138: 53–73. [CrossRef] McNeil, Alexander J., and Johanna Nešlehová. 2009. Multivariate Archimedean copulas, d -monotone functions and l1 -norm symmetric distributions. The Annals of Statistics 37: 3059–97. [CrossRef] Nelsen, Roger B. 2006. An Introduction to Copulas. Berlin/Heidelberg: Springer. Quessy, Jean-François, and Martin Durocher. 2019. The class of copulas arising from squared distributions: Properties and inference. Econometrics and Statistics 12: 148–66. [CrossRef]
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