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Checkerboard copula defined by sums of random variables. (English) Zbl 1437.62181
Summary: We consider the problem of finding checkerboard copulas for modeling multivariate distributions. A checkerboard copula is a distribution with a corresponding density defined almost everywhere by a step function on an \(m\)-uniform subdivision of the unit hyper-cube. We develop optimization procedures for finding copulas defined by multiply-stochastic matrices matching available information. Two types of information are used for building copulas: 1) Spearman rho rank correlation coefficients; 2) Empirical distributions of sums of random variables combined with empirical marginal probability distributions. To construct checkerboard copulas we solved optimization problems. The first problem maximizes entropy with constraints on Spearman rho coefficients. The second problem minimizes some error function to match available data. We conducted a case study illustrating the application of the developed methodology using property and casualty insurance data. The optimization problems were numerically solved with the AORDA Portfolio Safeguard (PSG) package, which has precoded entropy and error functions. Case study data, codes, and results are posted at the web.
MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H12 Estimation in multivariate analysis
90C25 Convex programming
91-10 Mathematical modeling or simulation for problems pertaining to game theory, economics, and finance
62P20 Applications of statistics to economics
Software:
PSG
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References:
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