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Bivariate copulas with quadratic sections. (English) Zbl 0857.62060

Summary: We present a method for constructing bivariate copulas with quadratic sections. These copulas are derived from simple univariate real-valued functions on the interval \([0,1]\). As a consequence, various positive dependence properties (such as quadrant dependence and total positivity), measures of association (such as Spearman’s rho and Kendall’s tau), stochastic orderings, and various notions of symmetry are shown to be equivalent to certain simple properties of the above-mentioned univariate functions. Several examples, some of which can be viewed as generalizations of the Farlie-Gumbel-Morgenstern family, are presented to illustrate the facility with which these copulas can be constructed.

MSC:

62H20 Measures of association (correlation, canonical correlation, etc.)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
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