Nonparametric Inference for Mathai–Haubold Extropy: Asymptotic Results and an Associated Cressie–Read-Type Divergence

Source abstract

Extropy, the complementary dual of Shannon entropy, has Rényi and Tsallis generalizations. For finite discrete distributions, a general asymptotic theory for plug-in estimators of functionals F(p)=∑iϕ(pi) is available in an arXiv preprint. Independently, the Mathai–Haubold entropy is a power-kernel, one-parameter generalization of Shannon’s differential entropy that recovers Shannon and Tsallis entropies as special or limiting cases. Its cumulative version, based on the distribution and survival functions rather than the density, has been studied through series representations involving moments of extreme order statistics. In this paper, We apply the Mathai–Haubold power kernel, so far used for cumulative entropies of continuous lifetimes, to the complementary probabilities 1−pi and study the resulting Mathai–Haubold extropy JαMH(p). We show that JαMH is an exact affine transform of the power sum S2−α(p)=∑i(1−pi)2−α that also defines the Tsallis extropy. It is therefore a reparameterization of the Tsallis extropy at the complementary order 2−α, not an independent measure. We also study the associated divergence DαMH(p∥q). It is not a new class of divergence: it equals (r−1) times a Cressie–Read (Csiszár f-) divergence between the normalized complementary probability vectors. Hence, it is nonnegative and jointly convex, and it satisfies a data-processing inequality for Markov processing of those normalized vectors; we show by an explicit counterexample that this does not extend to arbitrary coarsening of the original categorical variable. We prove asymptotic normality of its plug-in estimator for a fixed reference distribution q. A Monte Carlo study confirms n-consistency and O(n−1) bias, and adequate normal approximation of the extropy estimator from n≳200. An application to observed categorical data (cut grades of 53,940 diamonds) illustrates the inference; a warranty-claim example is hypothetical and serves only as an illustration.