On general weighted cumulative residual (past) extropy of extreme order statistics

Abstract: Weighted extropy has recently emerged as a flexible information measure for quantifying uncertainty, with particular relevance to order statistics. In this paper, we introduce and study a weighted cumulative analogue of extropy, extending the framework of weighted cumulative residual and cumulative past entropies to extreme order statistics. Specifically, we define the general weighted cumulative residual extropy (GWCREx) for the smallest order statistic and the general weighted cumulative past extropy (GWCPEx) for the largest order statistic, along with their dynamic versions. We show that these weighted measures and their dynamic counterparts uniquely characterize the underlying distribution. Moreover, we establish new characterization results for two widely used reliability models: the generalized Pareto distribution and the power distribution. The proposed framework provides a unified information-theoretic tool for analysing extreme lifetimes in reliability engineering and survival analysis.

• The paper introduces weighted cumulative residual and past extropy measures that extend uncertainty quantification in extreme order statistics.
• It derives explicit analytic expressions, monotonicity theorems, and bounds for various lifetime distributions, aiding system reliability analysis.
• The work offers new stochastic ordering results and unique distribution characterizations, bridging theoretical insights with practical risk evaluation.

General Weighted Cumulative Residual/Past Extropy of Extreme Order Statistics

Introduction and Motivation

This work provides a comprehensive extension and unification of extropy-based uncertainty measures, with a primary focus on extreme order statistics. It specifically constructs general weighted cumulative residual extropy (GWCREx) for the minimum order statistic and general weighted cumulative past extropy (GWCPEx) for the maximum, alongside their dynamic variants. Building on the duality between entropy and extropy [Lad et al., 2015] and recent developments in weighted and cumulative information measures [Balakrishnan et al., 2020; Jahanshahi et al., 2020], this paper systematically incorporates flexible weighting, order statistics, and time-localization to formulate robust quantifiers of distributional uncertainty. The weighting mechanism enables practitioners to focus on regions of practical interest (e.g., heavy tails, high-risk ages) and to integrate risk or cost-based criteria.

Order statistics, especially extremes, play pivotal roles in reliability analysis and survival modeling, where minimum/largest order statistics describe weakest-link (series) and strongest-link (parallel) systems, respectively. Prior extropy-based measures lacked cumulative or weight-adjusted forms, particularly for such statistics, limiting fine-grained system analysis and characterization. This paper fills this gap and establishes strong theoretical properties, including distributional identifiability, explicit distributional forms for common models, and distribution characterizations.

Definitions and Framework

The paper generalizes extropy measures by considering integrals of squared survival or distribution functions, modulated by user-defined weight functions. For a non-negative absolutely continuous random variable $X$ with cdf $F$, sf $\overline{F}$, and pdf $f$, and a weight function $w$, the following are defined:

• GWCREx (Smallest Order Statistic $X_{1:n}$):

$$\xi_J^w(X_{1:n}) = -\frac{1}{2} \int_0^\infty w(x) [\overline{F}(x)]^{2n} dx$$

This quantifies information "left out" by the rarest minimum lifetimes, with the weight adapting the measure.

• GWCPEx (Largest Order Statistic $X_{n:n}$):

$$\overline{\xi}_J^w(X_{n:n}) = -\frac{1}{2} \int_0^\infty w(x) [F(x)]^{2n} dx$$

• Dynamic Versions (Conditional on Survival or Failure to $t$):

$F$0

captures residual uncertainty given survival to $F$1, while

$F$2

quantifies past uncertainty given failure prior to $F$3.

This differential and cumulative structure is a significant technical advancement, encoding not only overall but also time-localized and regionally-weighted uncertainty of extreme samples in analytical form.

Analytical Results and Properties

Explicit Expressions and Bounds

The paper provides closed-form and semi-closed expressions for GWCREx/GWCPEx for major lifetime distributions, including:

• Uniform, power, Weibull, folded Cramér, Pareto, and generalized Pareto distributions, with varied $F$4 (constant, monomial, logarithmic), using quantile and transformation techniques.
• Key results show that, for $F$5, the measure yields analytic forms involving Beta or Gamma functions related to the model's parameters.

Monotonicity and Inequality Relations

Rigorous monotonicity theorems are established:

• GWCREx is strictly increasing in the sample size $F$6 for all non-negative $F$7 (i.e., larger systems exhibit greater minimal-lifetime uncertainty).
• $F$8, reflecting the increased uncertainty of series system failure times relative to single components.
• For GWCPEx, analogous monotonicity and upper bound results are shown for the maximum order statistic.

Lower bounds are provided, e.g., $F$9 and $\overline{F}$0, where $\overline{F}$1 is a weighted mean and $\overline{F}$2 the mean residual life. These allow practical, moment-based uncertainty assessment without full distribution knowledge.

Stochastic Ordering and Invariance

Definitions of weighted dynamic cumulative residual/past extropy stochastic orders are introduced:

• $\overline{F}$3 if $\overline{F}$4 for all $\overline{F}$5.
• Invariance under affine transformations and shift-scale families is formalized, with explicit transformation behavior for both $\overline{F}$6 and the associated $\overline{F}$7 mappings.

Distribution Characterization and Identifiability

The most significant theoretical result is the unique characterization of the parent distribution via GWCREx (and GWCPEx) across all order statistics, under a Műntz–Szász sequence condition:

• For a strictly increasing sequence $\overline{F}$8 with $\overline{F}$9 and translation-invariant $f$0, equality of GWCREx for all $f$1 determines the law up to location (or location–scale) shifts. The same principle holds for GWCPEx and maximal order statistics.

Further, the paper gives new, low-order characterization theorems for generalized Pareto and power distributions in terms of constancy of specific functional forms involving GWCREx/GWCPEx and the mean residual/inactivity functions. The derivative conditions imposed on these extropy measures (i.e., their constancy in $f$2) uniquely identify important reliability families, bridging information theory and classical parametric modeling.

Dynamic and Weighted Extensions

The extension to dynamic (i.e., conditional-on-survival or failure) and weighted forms is of particular applied relevance. The paper’s dynamic results include:

• New bounds, monotonicity, and functional inequalities for weighted dynamic extropy.
• Ordering results (e.g., with increasing hazard/reversed hazard, the system uncertainty ordering is preserved).
• Sensitivity analyses, showing how weight/importance choices (e.g., emphasizing high-cost failures via $f$3) alter the quantification of dynamic uncertainty.

Implications and Applications

The technical implications are both theoretical and practical:

• Identifiability: The ability to uniquely characterize standard families via order-statistic extropy measures positions these tools as strong alternatives or complements to entropy and classical likelihood-based identification methods.
• Reliability Engineering/Survival Analysis: Practitioners can tailor uncertainty quantification to real-world cost, risk, or reliability priorities by adapting $f$4, and the formulas admit analytic/numerical evaluation for a wide class of models.
• Statistical Inference: The framework facilitates inference of model parameters or family membership from sample order statistics, especially in censored or incomplete data scenarios. The moment-type bounds directly translate to estimation procedures.
• Stochastic Ordering: The introduction of WDCREx and DCPWEx orders translates to new system reliability comparison tools, with invariance properties allowing portability across transformed/standardized systems.

Empirical Illustration

The real-data demonstration on the Boeing 720 airconditioning failure dataset exemplifies calculation and interpretation. Empirical GWCREx values for $f$5 (with constant and linear weights) reveal that increasing $f$6 leads to less negative extropy, indicating increased uncertainty and justifying the theoretical predictions. Weighting by $f$7 boosts the magnitudes, underlining sensitivity to late-occurring, high-impact failures.

Conclusion

The paper establishes a unifying, general, and flexible framework for quantifying and analyzing uncertainty in the context of extreme order statistics. By integrating extropy with weighting, cumulation, and time-dynamics, it both advances theoretical information theory and delivers practical diagnostic and modeling tools for reliability and survival analytics. The strong identifiability and distributional characterization properties represent robust theoretical guarantees, and the modular weighting scheme ensures relevance for a wide range of applications.

Future directions logically include:

• Development of robust estimation and inference techniques for GWCREx/GWCPEx in complex or high-dimensional systems.
• Integration of these measures with learning algorithms, e.g., for model selection, uncertainty quantification in risk-sensitive ML, or anomaly detection.
• Exploration of more general dependence structures (e.g., via copulas) and extension to random censoring or truncation models.

The presented results set the stage for extropy-based analytics to take a central role in both parametric and nonparametric modeling of extremal events and order-dependent risk.