Bivariate copulas with quadratic sections

Abstract

Abstract In this paper 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. Keywords: Copulasfamilies of bivariate copulasmeasures of dependenceconcepts of dependencestochastic orderingsradial symmetryAMS 199I subject classifications: Primary 62HO5secondary 62H20