Weighted Entropy & Information Generating Functions

• Weighted entropy and information generating functions are mathematical constructs that extend classical entropy by incorporating weight factors to better quantify uncertainty.
• They exhibit robust properties like monotonicity, convexity, and unique distributional characterizations, making them essential for stochastic ordering and reliability analyses.
• These frameworks facilitate practical applications including system reliability assessment, lifetime modeling, and hypothesis testing via nonparametric estimation and dynamic forms.

Weighted entropy and information generating functions are central objects in modern information theory and reliability analysis, providing versatile frameworks for quantifying uncertainty, variability, and information content in both discrete and continuous probability models. These constructs generalize classical entropic measures by introducing weighting (typically via auxiliary functions or tail probabilities) and generating structures that encapsulate a broad spectrum of entropic and variability characteristics. Recent advancements have formalized a wide variety of weighted entropy generating functions (WEGF), weighted cumulative residual entropy generating functions (WCREGF), their dynamic forms, and cumulative information generating function generalizations, leading to new tools for model characterization, stochastic ordering, testing, and system reliability.

1. Foundational Definitions and General Frameworks

Weighted entropy generating functions (WEGF) and their extensions are designed to recover weighted entropy metrics as derivatives or parameter limits and to support a broader array of structural and inferential analyses.

For a nonnegative absolutely continuous random variable $X$ with density $f$, distribution $F$, and survival function $\bar F$, several core definitions have emerged:

• Weighted Entropy Generating Function (WEGF):

$$B_s(W, X) = \int_0^\infty x\, f(x)^s\,dx = \mathbb{E}[X\,f(X)^{s-1}]$$

with $s \geq 0, s \neq 1$, where

$$\left.\frac{d}{ds}B_s(W, X)\right|_{s=1} = -H^w(X), \quad H^w(X) = -\int x\,f(x)\ln f(x)\,dx$$

gives the weighted Shannon entropy (S. et al., 20 Jul 2025).

• General Weighted Information Generating Function (GWIGF):

$$I^{\omega}_\beta(X) = \int_0^\infty \omega(x)\,f(x)^\beta\,dx$$

for a measurable weight function $\omega(x) \ge 0$ and parameter $\beta \geq 1$. Its expansion:

$$I^{\omega}_\beta(X) = \sum_{k=0}^{\infty} \frac{(1-\beta)^k}{k!} H^{\omega}_{k}(X)$$

connects it to the family of weighted Shannon entropies and their moments (Saha et al., 2023).

• Weighted Cumulative Residual Entropy Generating Function (WCREGF):

$C_s^W(F) = \int_0^\infty x\,\bar F(x)^s\,dx$

and its dynamic version,

$$C_s^W(F; t) = \int_t^\infty x\,\left[\frac{\bar F(x)}{\bar F(t)}\right]^s\,dx$$

These functionals can recover the weighted cumulative residual entropy (WCRE) through derivatives at $s=1$ (S. et al., 9 Feb 2024).

• Discrete Weighted Information Generating Function:

For a discrete probability distribution $P = (p_i)$ with associated weights $w_i$,

$G_w(s) = \sum_{i} p_i^{1 - w_i(1-s)}$

recovers the weighted Shannon entropy and links to Rényi and Tsallis entropy when the weights are constant (Srivastava et al., 2015).

2. Properties and Monotonicity

Weighted entropy generating functions exhibit robust analytic and stochastic properties, which support their use as information-theoretic and statistical tools.

• Monotonicity and Convexity: For suitable parameter regimes (e.g., $s \geq 1$), WEGFs and WCREGFs are non-increasing (or non-decreasing) and strictly convex in the generating parameter due to the non-negativity of terms involving logs of probabilities or densities. The second derivative is strictly positive (e.g., for continuous $X$, $\frac{\partial^2}{\partial s^2}C_s^W(F) > 0$), supporting convex ordering (S. et al., 9 Feb 2024, Srivastava et al., 2015).
• Normalization and Shift-Dependence: The GWIGF is shift-dependent (i.e., affected by affine transformations of the argument), with explicit scaling formulas. For example, under affine transformation $Y = aX + b, a > 0$, for WEGF:

$$B_s(W, Y) = a^{1-s}\bigl[ a B_s(W, X) + b B_s(X) \bigr]$$

(S. et al., 20 Jul 2025, Saha et al., 2023).

• Product Structure: For independent random variables, WEGFs factorize, e.g., $B_s(W, (X, Y)) = B_s(W, X)B_s(W, Y)$ (S. et al., 20 Jul 2025).
• Stochastic Orders and Variability: Monotonicity properties of the dynamic forms (DWCREGF, WREGF) motivate stochastic orderings such as "IWREGF" (increasing WREGF) and "DWREGF" (decreasing WREGF), defining classes of distributions and supporting comparison under dispersive order (S. et al., 20 Jul 2025).

3. Characterization and Uniqueness Theorems

Weighted entropy generating functions can uniquely characterize underlying probability distributions and provide necessary and sufficient conditions associated with constant functionals.

• Distributional Uniqueness: The mapping $t \mapsto C_s^W(F; t)$ uniquely determines the survival function $\bar F(t)$ (S. et al., 9 Feb 2024).
• Characterization of Classical Families:
  • For the DWCREGF, $C_s^W(F; t)$ is constant in $t$ if and only if $X$ is Rayleigh, with $h(t) = t/(s k)$, reflecting a quadratic hazard (S. et al., 9 Feb 2024).
  • For the WREGF, constancy in $t$ provides characterizations for Weibull and Pareto laws: for $s \neq 2$, relate to Weibull $(\text{shape}=s/(s-1))$; for $s=2$, characterize Pareto type I (S. et al., 20 Jul 2025).
• Hazard Rate and Mean Residual Life Linkages: Differentiating the dynamic forms with respect to $t$ yields explicit hazard rate identities, e.g.,

$$h(t) = \frac{t + \partial_t C_s^{W}(F; t)}{s C_s^{W}(F; t)}$$

or in terms of mean residual life for the Rayleigh family (S. et al., 9 Feb 2024, S. et al., 20 Jul 2025).

4. Weighted Cumulative and Residual Entropy Generating Families

The extension to cumulative, mixture, and residual settings generalizes the entropic and information-theoretic landscape:

• Cumulative Information Generating Function (CIGF):

$$G_X(\alpha, \beta) = \int_{l}^{r} F(x)^{\alpha} \bar F(x)^{\beta} dx$$

underlies a two-parameter family encompassing Gini mean differences, cumulative residual entropy (CRE), and their higher-order and fractional analogues through differentiation with respect to $\alpha$ or $\beta$ (Capaldo et al., 2023).

• Weighted and Distorted CIGFs: Introducing mixture or distortion functions $q_i$ extends these functionals, yielding

$$\hat G_X(q_1, q_2; F_T) = \int F_{q_1}(x) \bar F_{q_2}(x)\,dF_T(x)$$

supporting stress-strength, order-statistic, and multivariate reliability models (Capaldo et al., 2023).

• Residual Forms: Both GWIGF and WREGF have residual analogues for the residual lifetime at time $t$, crucial in reliability and survival analysis. Estimation approaches employ empirical or kernel estimation for practical use (S. et al., 20 Jul 2025, Saha et al., 2023).

5. Statistical Testing and Model Checking

Weighted entropy generating functions serve as the backbone for new hypothesis testing procedures exploiting distributional characterizations.

• Goodness-of-Fit for Rayleigh Law (DWCREGF): Utilizing the constancy property, the test statistic

$$A(F) = \int_{0}^{\infty} [ t\,\bar F(t)^{s+1} - s \int_t^\infty x\,\bar F(x)^s\,dx ]\,dt$$

simplifies to the difference of expectations of minimum squared order statistics, enabling a U-statistic-based detector with demonstrated power outperforming Kolmogorov–Smirnov and related competitors in various scenarios (S. et al., 9 Feb 2024).

• Goodness-of-Fit for Pareto I (WREGF): The plug-in statistic

$$\Delta(F) = \int_1^\infty \{ 3xF(x) - x \} f(x)^2 dx$$

is zero if and only if $X$ is Pareto I, supporting a nonparametric kernel-based test with finite-sample accuracy shown via Monte Carlo (S. et al., 20 Jul 2025).

6. Connections to Classical and Generalized Entropic Measures

Weighted entropy generating functions both unify and extend classical entropic architectures.

• Recovery of Classical Entropies: Derivatives at special values retrieve cumulative entropy, cumulative residual entropy (CRE), weighted Shannon entropy, and higher/fractional variants (Saha et al., 2023, Capaldo et al., 2023, Srivastava et al., 2015).
• Generalization to Tsallis and Rényi Families: Under uniform weights in the discrete case, the weighted IGF recovers Rényi and Tsallis entropy forms by appropriate parameter choices (Srivastava et al., 2015).
• Cumulative/Gini Interpretation: CIGFs and their weighted extensions generalize Gini mean difference and supply continuous, dispersion-based variability orders applicable to actuarial science and risk management (Capaldo et al., 2023).

7. Applications and Computational Considerations

Weighted entropy generating functions are instrumental in reliability theory, statistical inference, and risk analysis.

• Reliability and Lifetime Modeling: WCREGF and WREGF provide age-dependent entropy curves for lifetimes, enabling freshness of information, model checks, and risk ordering (S. et al., 9 Feb 2024, S. et al., 20 Jul 2025).
• Estimation: Non-parametric and kernel-smoothed density estimators support inference for weighted generating functionals and their derivative-based statistics, with empirical bias and MSE assessments guiding applied practice (S. et al., 20 Jul 2025, Saha et al., 2023).
• Extended System Reliability: The cumulative and weighted frameworks support stress-strength, $k$-out-of-$n$ systems, and multivariate extensions relevant in multi-component system analysis and actuarial contexts (Capaldo et al., 2023).

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In summary, the landscape of weighted entropy and information generating functions comprises a comprehensive and extensible toolkit for the measurement and analysis of uncertainty, reliability, and information, with well-developed theoretical grounding, distributional characterizations, inferential techniques, and demonstrated empirical merits in simulation and real-data contexts (S. et al., 9 Feb 2024, S. et al., 20 Jul 2025, Saha et al., 2023, Capaldo et al., 2023, Srivastava et al., 2015).