A New Measure of Monotone Dependence by Using Sobolev Norms for Copula

Abstract

Dependence structure, e.g. measures of dependence, is one of the main studies in correlation analysis. In [10], B. Schweizer and E.F. Wolff used L$^{p}$-metric $d_{L^{p}}(C,P)$ to obtain a measure of monotone dependence where P is the product copula or independent copula, and in [11] P. A. Stoimenov defined Sobolev metric $d_{S}(C,P)$ to construct the measure $\omega (C)$ for a class of Mutual Complete Dependences (MCDs). Due to the fact that the class of monotone dependence is contained in the class of MCDs, we constructed a new measure of monotone dependence, $\lambda (C),$ based on Sobolev metric which can be used to characterize comonotonic, countermonotonic and independence.