On some entropy and divergence type measures of variability and dependence for mixed continuous and discrete variables

It is well known the role of entropies and divergences in statistics and related fields as indices of the diversity or variability and pseudodistances between statistical populations, as well. The definition of these measures is extended in the case of mixed continuous and discrete variables, a case which is common in practice in several fields in science and engineering. Two descriptive measures are proposed, in this paper, for mixed, continuous and discrete multivariate data. Some properties of the measures are clarified and the asymptotic behavior of one of them is outlined when it is applied in the elliptic family of multivariate distributions. Explicit expressions of the measures are derived in the appendix for specific elliptic distributions.