Mean-square distance of information contents

We introduce a functional that quantifies discrepancies in an $L^2$-sense between the marginal information contents of two absolutely continuous random variables. The proposed measure is symmetric, nonnegative, and computationally tractable through its connection with entropy and varentropy, and the availability of closed forms in specific parametric models. Its analytical properties are investigated, using exchangeability arguments and by analyzing the effect of separable transformations. Upper and lower bounds are derived under a new notion of weak density dependence and the quadrant dependence. A new relative mean information inconsistency measure is introduced, leading to applications to the proportional hazard rate model and order statistics. From a statistical perspective, non-parametric kernel-based estimators are constructed and their convergence properties are established. Simulation studies illustrate finite-sample performance in terms of bias and mean squared error. The methodology is further illustrated through an application to a real dataset.