Characterizations of Multivariate Implicit Dependence Copulas

TheSantiwipanont, T.Sumetkijakan, S.Yanpaisan, N. copula C of continuously distributed random variables \(X_1,\dots ,X_d\) is said to be an implicit dependence copula if there are Borel functions \(\alpha _1,\dots ,\alpha _d\) such that \(\alpha _1(X_1),\dots ,\alpha _d(X_d)\) are equal almost surely and continuously distributed, that is their common distribution function is continuous. Bivariate implicit dependence copulas have recently been characterized in terms of a generalized Markov product. In this manuscript, the characterizations are extended to the multivariate case in terms of a product of d copulas, called \(\mathscr {A}\) -product where \(\mathscr {A}\) is a class of copulas \(A_t\) , \(t\in [0,1]\) . The class of implicit dependence d-copulas are characterized as \(\mathscr {A}\) -products of d complete dependence copulas. Explicit forms of the joining copulas \(A_t\) are obtained when the functions \(\alpha _i\) are countably piecewise monotonic surjections.