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COMPATIBILITY AND ATTAINABILITY OF MATRICES OF CORRELATION-BASED MEASURES OF CONCORDANCE

Published online by Cambridge University Press:  21 May 2019

Marius Hofert  and
Takaaki Koike
Marius Hofert*
Affiliation:
Department of Statistics and Actuarial Science University of WaterlooWaterloo, ON, Canada
Takaaki Koike
Affiliation:
Department of Statistics and Actuarial Science University of WaterlooWaterloo, ON, Canada E-mail: [email protected]
*
E-mail: [email protected]
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Abstract

Measures of concordance have been widely used in insurance and risk management to summarize nonlinear dependence among risks modeled by random variables, which Pearson’s correlation coefficient cannot capture. However, popular measures of concordance, such as Spearman’s rho and Blomqvist’s beta, appear as classical correlations of transformed random variables. We characterize a whole class of such concordance measures arising from correlations of transformed random variables, which includes Spearman’s rho, Blomqvist’s beta and van der Waerden’s coefficient as special cases. Compatibility and attainability of square matrices with entries given by such measures are studied—that is, whether a given square matrix of such measures of concordance can be realized for some random vector and how such a random vector can be constructed. Compatibility and attainability of block matrices and hierarchical matrices are also studied due to their practical importance in insurance and risk management. In particular, a subclass of attainable block Spearman’s rho matrices is proposed to compensate for the drawback that Spearman’s rho matrices are in general not attainable for dimensions larger than three. Another result concerns a novel analytical form of the Cholesky factor of block matrices which allows one, for example, to construct random vectors with given block matrices of van der Waerden’s coefficient.

Keywords

Tansformed rank correlation coefficientsmatrices of pairwise measures of concordancecompatibilityattainabilitycopula modelsCholesky factorblock correlation matriceshierarchical matrices
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 3 , September 2019 , pp. 885 - 918
Copyright
© Astin Bulletin 2019 

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References

Blomqvist, N. (1950) On a measure of dependence between two random variables. The Annals of Mathematical Statistics, 21(4), 593–600.CrossRefGoogle Scholar
Carley, H. and Taylor, M.D. (2002) A new proof of Sklar’s theorem. In Distributions with Given Marginals and Statistical Modelling, pp. 29–34, Dordrecht: Springer.CrossRefGoogle Scholar
Devroye, L. and Letac, G. (2015) Copulas with prescribed correlation matrix. In Memoriam Marc Yor-Séminaire de Probabilités XLVII, pp. 585–601, Switzerland: Springer.CrossRefGoogle Scholar
Edwards, H.H. and Taylor, M.D. (2009) Characterizations of degree one bivariate measures of concordance. Journal of Multivariate Analysis, 100(8), 1777–1791.CrossRefGoogle Scholar
Edwards, H.H., Mikusiński, P. and Taylor, M.D. (2004) Measures of concordance determined by d4-invariant copulas. International Journal of Mathematics and Mathematical Sciences, 2004(70), 3867–3875.CrossRefGoogle Scholar
Edwards, H.H., Mikusiński, P. and Taylor, M. (2005) Measures of concordance determined by d4-invariant measures on (0.1)2. Proceedings of the American Mathematical Society, 133(5), 1505–1513.CrossRefGoogle Scholar
Embrechts, P. and Hofert, M. (2013) A note on generalized inverses. Mathematical Methods of Operations Research, 77(3), 423–432, doi:10.1007/s00186-013-0436-7.CrossRefGoogle Scholar
Embrechts, P., McNeil, A.J. and Straumann, D. (1999) Correlation: Pitfalls and alternatives. RISK Magazine, pp. 69–71.Google Scholar
Embrechts, P., McNeil, A.J. and Straumann, D. (2002) Correlation and Dependency in Risk Management: Properties and pitfalls. In Risk Management: Value at Risk and Beyond (ed. Dempster, M.), pp. 176–223. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Embrechts, P., Hofert, M. and Wang, R. (2016) Bernoulli and tail-dependence compatibility. The Annals of Applied Probability, 26(3), 1636–1658, doi: 10.1214/15-AAP1128.CrossRefGoogle Scholar
Genest, C., Gendron, M. and Bourdeau-Brien, M. (2009) The advent of copulas in finance. The European Journal of Finance, 15, 609–618.CrossRefGoogle Scholar
Górecki, J., Hofert, M. and Holeňa, M. (2017) On structure, family and parameter estimation of hierarchical Archimedean copulas. Journal of Statistical Computation and Simulation, 87(17), 3261–3324, doi:10.1080/00949655.2017.1365148.CrossRefGoogle Scholar
Hofert, M. (2010) Sampling Nested Archimedean Copulas with Applications to CDO Pricing, PhD thesis, Südwestdeutscher Verlag für Hochschulschriften AG & Co. KG, ISBN 978-3-8381-1656-3.Google Scholar
Hofert, M. (2011) Efficiently sampling nested Archimedean copulas. Computational Statistics & Data Analysis, 55, 57–70, doi:10.1016/j.csda.2010.04.025.CrossRefGoogle Scholar
Hofert, M. (2012) A stochastic representation and sampling algorithm for nested Archimedean copulas. Journal of Statistical Computation and Simulation, 82(9), 1239–1255, doi:10.1080/00949655.2011.574632.CrossRefGoogle Scholar
Hofert, M. and Scherer, M. (2011) CDO pricing with nested Archimedean copulas. Quantitative Finance, 11(5), 775–787, doi:10.1080/14697680903508479.CrossRefGoogle Scholar
Huang, J. and Yang, L. (2010) Correlation matrix with block structure and efficient sampling methods. Journal of Computational Finance, 14(1), 81.CrossRefGoogle Scholar
Huber, M. and Maric, N. (2015) Multivariate distributions with fixed marginals and correlations. Journal of Applied Probability, 52(2), 602–608.CrossRefGoogle Scholar
Huber, M. and Maric, N. (2017) Bernoulli correlations and cut polytopes, https://arxiv.org/abs/1706.06182.Google Scholar
Hürlimann, W. (2012) On trivariate copulas with bivariate linear spearman marginal copulas. Journal of Mathematics and System Science, 2(6), 368–383.Google Scholar
Hürlimann, W. (2014) A closed-form universal trivariate pair-copula. Journal of Statistical Distributions and Applications, 1(1), 7.CrossRefGoogle Scholar
Jaworski, P., Durante, F., Härdle, W.K. and Rychlik, T., eds. (2010) Copula Theory and Its Applications, vol. 198, Lecture Notes in Statistics – Proceedings, Springer.CrossRefGoogle Scholar
Joe, H. (1990) Multivariate concordance. Journal of Multivariate Analysis, 35(1), 12–30.CrossRefGoogle Scholar
Joe, H. (1996) Families of m-variate distributions with given margins and m(m − 1)/2 bivariate dependence parameters. In Distributions with fixed marginals and related topics (eds. Rüschendorf, L., Schweizer, B., and Taylor, M. D.), pp. 120–141. Hayward, CA: Institute of Mathematical Statistics.CrossRefGoogle Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. Dordrecht: Chapman & Hall/CRC.CrossRefGoogle Scholar
Kimberling, C.H. (1974) A probabilistic interpretation of complete monotonicity. Aequationes Mathematicae, 10, 152–164.CrossRefGoogle Scholar
Kurowicka, D. and Cooke, R. (2001) Conditional, partial and rank correlation for the elliptical copula; dependence modelling in uncertainty analysis. Proceedings ESREL, pp. 1795–1802.Google Scholar
Laurent, M. and Poljak, S. (1995) On a positive semidefinite relaxation of the cut polytope. Linear Algebra and its Applications, 223, 439–461.CrossRefGoogle Scholar
McNeil, A.J. (2008) Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78(6), 567–581.CrossRefGoogle Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2015) Quantitative Risk Management: Concepts, Techniques, Tools, 2nd ed., Princeton, NJ: Princeton University Press.Google Scholar
Mosteller, F. (2006) On some useful “inefficient” statistics. In Selected Papers of Frederick Mosteller (eds Fienberg, S.E. and Hoaglin, D.C.), pp. 69–100, New York: Springer.CrossRefGoogle Scholar
Müller, A. and Scarsini, M. (2000) Some remarks on the supermodular order. Journal of Multivariate Analysis, 73, 107–119.CrossRefGoogle Scholar
Murdoch, D.J., Craiu, R.V. and Meng, X.L. (2001) On the edge: Statistics & computing: Chance and fractals. Chance, 14(2), 47–52.Google Scholar
Nelsen, R.B. (2006) An Introduction to Copulas. New York: Springer-Verlag.Google Scholar
Pitowsky, I. (1991) Correlation polytopes: Their geometry and complexity, Mathematical Programming, 50(1–3), 395–414.CrossRefGoogle Scholar
Raymaekers, J. and Rousseeuw, P.J. (2018) Fast robust correlation for high dimensional data, https://arxiv.org/abs/1712.05151.Google Scholar
Roustant, O. and Deville, Y. (2017) On the validity of parametric block correlation matrices with constant within and between group correlations, https://arxiv.org/abs/1705.09793.Google Scholar
Scarsini, M. (1984) On measures of concordance, Stochastica, 8(3), 201–218.Google Scholar
Sidak, Z., Sen, P.K. and Hajek, J. (1999) Theory of rank tests. San Diego, CA: Academic Press.Google Scholar
Spearman, C. (1904) “General Intelligence,” objectively determined and measured, The American Journal of Psychology, 15(2), 201–292.CrossRefGoogle Scholar
Tchen, A.H. (1980) Inequalities for distributions with given marginals. The Annals of Probability, 8(4), 814–827.CrossRefGoogle Scholar
Tropp, J.A. (2018) Simplicial faces of the set of correlation matrices, Discrete & Computational Geometry, 60(2), 512–529.CrossRefGoogle Scholar
Wang, B., Wang, R. and Wang, Y. (2018) Compatible matrices of spearman’s rank correlation, http://arxiv.org/abs/1810.03477.Google Scholar