Cumulative Generalized Entropies of Mathai–Haubold Type and Moment Functions of Order Statistics

The cumulative Mathai–Haubold residual entropy (CMHRE) and cumulative Mathai–Haubold entropy (CMHCE) are introduced as one-parameter extensions of cumulative residual entropy (CRE) and cumulative entropy (CE), respectively. By expanding the integrands using a generalized binomial series and applying moment identities for order statistics, both measures are represented as absolutely convergent weighted sums involving differences between the expected maxima and minima of independent and identically distributed samples. Generalized weighted versions are also developed using an arbitrary nonnegative weight function, leading to representations in terms of weighted moments of extreme order statistics. Closed-form expressions are obtained for exponential and uniform distributions, while additional examples demonstrate the effects of power and logarithmic weight functions. Three application oriented examples, involving an excess-of-loss insurance layer, a quadratic warranty-cost weight, and a discounted maintenance-cost weight, further illustrate the practical relevance of the proposed measures. A comparison with PDF-based weighted entropies, together with numerical and graphical analyses, is also provided.