On Weighted Mathai-Haubold Entropy Measures

1 Introduction

• Residual entropy targets remaining-lifetime uncertainty after an item has survived to time t, where ordinary Shannon entropy does not provide that information.The introduction also describes past entropy in terms of the lifetime before t, conditional on the item having failed by t.Source: §1 · pp. 1–2
• Mathai-Haubold entropy generalizes Shannon entropy and reduces to it as α approaches 1; α is described as a flexibility parameter.The introduction also identifies earlier Mathai-Haubold residual and past entropy measures.Source: §1 · p. 2
• Weighted entropy addresses the omission of realized values by assigning non-negative utility weights to events; the linear weight x gives larger values more importance.The introduction contrasts this with entropy measures that consider only probabilistic information.Source: §1 · p. 2
• The paper introduces weighted Mathai-Haubold entropy and its residual and past versions, then develops aging classes based on the weighted residual and past measures.It also studies properties and inequalities, and introduces non-parametric estimators for independent and dependent data.Source: §1 · pp. 2–3

2 Weighted Mathai-Haubold Entropy

• The section proposes weighted Mathai-Haubold entropy for a non-negative absolutely continuous random variable.Definition 2.1 specifies the measure for a non-negative continuous random variable with density g(x).Source: §2 · p. 3
• Table 1 presents WMHE for some distributions.Source: Table 1 · p. 3, §2 · p. 3
• An example compares shifted uniform random variables whose ordinary Mathai-Haubold entropies are equal.The passage introduces this example to illustrate WMHE, but the supplied text does not state the weighted-measure comparison.Source: §2 · p. 3
• The section examines how WMHE behaves under a linear transformation Y = aX + b, with a > 0 and b ≥ 0.Source: §2 · p. 4

3 Weighted Mathai-Haubold Residual Entropy

• WMHRE is introduced for a non-negative random variable, and the section reports calculations for some distributions.The calculations are presented in Table 2, though the supplied caption labels that table WMHPE.Source: §3 · p. 4, Table 2 · p. 4
• For finite, decreasing WMHRE with α ≠ 1 and 0 < α < 2, the measure uniquely determines the survival function.The proof proceeds through the hazard rate, which WMHRE uniquely determines.Source: §3 · p. 5
• When WMHRE is constant for all t and 1 < α < 2, the distribution is Weibull.The corresponding hazard function is also Weibull.Source: §3 · p. 6
• The section defines increasing and decreasing WMHRE aging classes by whether WMHRE is non-decreasing or non-increasing in t.The definitions name these classes IWMHRE(α) and DWMHRE(α).Source: §3 · p. 6

4 Weighted Mathai-Haubold Past Entropy

• WMHPE is introduced for a non-negative random variable, and the section reports calculations for some distributions.The calculations are presented in Table 3, whose supplied caption also labels it WMHPE.Source: §4 · p. 7, Table 3 · p. 7
• For finite, decreasing WMHPE with α ≠ 1 and 0 < α < 2, the measure uniquely determines the distribution function.The proof proceeds through the reversed hazard function, which WMHPE uniquely determines.Source: §4 · pp. 7–8
• The section defines increasing and decreasing WMHPE aging classes according to whether WMHPE is non-decreasing or non-increasing in t.These classes are named IWMHPE(α) and DWMHPE(α).Source: §4 · p. 8
• The section establishes a relationship among WMHE, WMHRE, and WMHPE, with a limiting approximation as α approaches 1.In that limit, the relation uses the survival and distribution functions as weights on the residual and past measures.Source: §4 · pp. 8–9

5 Some inequalities and bounds

• The section states lower bounds for WMHE, WMHRE, and WMHPE in terms of Shannon entropy.It also introduces inequalities for these three proposed measures.Source: §5 · p. 9
• For the residual measure, the section gives a bound involving residual entropy and the conditional log moment given X > t.The displayed inequality is stated for WMHRE.Source: §5 · p. 9
• The lower bound for WMHE is determined by a geometric mixture of the residual and past lower bounds.The section also gives the Shannon entropy decomposition into survival-weighted residual entropy, distribution-weighted past entropy, and two probability terms.Source: §5 · p. 10
• A unified Hardy-type inequality is developed for WMHE, WMHRE, and WMHPE.The section states the unified inequality for an interval I = [A, B] ⊆ [0, ∞].Source: §5 · p. 12

6 Nonparametric Estimation

• Kernel-based nonparametric estimators are proposed for weighted Mathai-Haubold entropy and its residual and past versions, with an extension to dependent data.The dependent-data extension is introduced after the estimators for independent data.Source: §6 · p. 13
• The estimators are consistent, as follows from consistency of the kernel density estimator.The section identifies the empirical survival estimator as the tail integral of the estimated density.Source: §6 · pp. 13, 15
• For dependent data, the section gives bias and variance expressions for density, distribution, and survival estimators under ρ-mixing conditions.The sequence is ρ-mixing when its mixing coefficient tends to zero as n tends to infinity.Source: §6 · pp. 13–14

7 Simulation Study

• The study uses 1000 samples from standard uniform and standard exponential reference distributions and evaluates bias and MSE at α = 0.10, 0.25, 0.50, and 0.75.Source: §7 · p. 16
• Bias and MSE of the weighted entropy estimator decrease as sample size increases, and lower α values give better estimates.The authors attribute sensitivity at α > 1 to large estimator errors from small errors in the density estimate, with the problem worsening as α approaches 2.Source: Table 4 · p. 16, Table 5 · p. 17, §7 · p. 19
• The residual and past entropy estimators work well in simulations from U(0,1) and Exp(1).The study reports their bias and MSE for different α and t values.Source: Table 6 · p. 17, Table 7 · p. 18, §7 · p. 19
• For dependent data, WMHRE and WMHPE show good performance in bias and MSE under an AR(1) model with ϕ = 0.5, α = 0.50, and Exp(1.5) errors.Source: §7 · p. 19, Table 8 · p. 19

• The paper proposes weighted Mathai-Haubold entropy and residual and past versions, studies their properties, develops bounds, constructs aging classes, and obtains a Weibull characterization result.Source: §8 · p. 19
• Nonparametric estimators are introduced for the three measures, their properties are studied, and their performance is evaluated by simulation.Source: §8 · p. 19
• The estimator discussion uses kernel-density methods for independent and dependent data; spacing- and quantile-based estimators remain possible directions for further work.Source: §8 · p. 19

• The work received no funding.Source: Funding · p. 20