Measuring stochastic dependence using ϕ-divergence

The problem of bivariate (multivariate) dependence has enjoyed the attention of researchers for over a century, since independence in the data is often a desired property. There exists a vast literature on measures of dependence, based mostly on the distance of the joint distribution of the data and the product of the marginal distributions, where the latter distribution assumes the property of independence. In this article, we explore measures of multivariate dependence based on the ϕ-divergence of the joint distribution of a random vector and the distribution that corresponds to independence of the components of the vector, the product of the marginals. Properties of these measures are also investigated and we employ and extend the axiomatic framework of Renyi [On measures of dependence, Acta Math. Acad. Sci. Hungar. 10 (1959) 441–451], in order to assert the importance of ϕ-divergence measures of dependence for a general convex function ϕ as well as special cases of ϕ. Moreover, we obtain point estimates as well as interval estimators when an elliptical distribution is used to model the data, based on ϕ-divergence via Monte Carlo methods.