Jensen-variance distance measure: a unified framework for statistical and information measures

The Jensen-variance (JV) distance measure is introduced and some properties are developed. The JV distance measure can be expressed using two interesting representations: the first one is based on mixture covariances, and the second one is in terms of the scaled variance of the absolute difference of two random variables. The connections between the JV distance measure and some well-known information measures, such as Fisher information, Gini mean difference, cumulative residual entropy, Fano factor, varentropy, varextropy, and chi-square distance measures, are examined. Specifically, the JV distance measure possesses metric properties and unifies most of the information measures within a general framework. It also includes variance and conditional variance as special cases. Furthermore, an extension of the JV distance measure in terms of transformed variables is provided. Finally, to demonstrate the usefulness of proposed methods, JV distance is applied to a real-life dataset related to fish condition factor index and some numerical results assuming skew-normal-distributed samples are presented.