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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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