Weighted Entropy Generating Function (WEGF)

• WEGF is a family of parameterized weighted power-integral constructions that generate entropy measures through differentiation and logarithmic transformations.
• It unifies classical and weighted entropies such as Shannon, Rényi, and residual entropies, and adapts to reliability and survival analysis.
• WEGF techniques enable practical statistical inference, model assessment, and characterization of lifetime distributions via explicit integral formulations.

Weighted Entropy Generating Function (WEGF) denotes a class of parameterized weighted power-integral constructions that generate entropy-type quantities through differentiation, logarithmic transformation, or both. In the most explicit usage, for a non-negative absolutely continuous random variable $X$ with pdf $f$, the WEGF is

$$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$

and its derivative at $s=1$ yields the weighted entropy $H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$ (S. et al., 20 Jul 2025). Closely related literatures use more general weights $\omega$ and define

$$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$

or interpret

$G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$

as the natural generating object behind weighted Rényi entropies (Saha et al., 2023, Sekeh, 2015). The term therefore refers not to a single universally fixed formula, but to a common generating principle connecting weighted Shannon entropy, weighted Rényi entropy, weighted residual entropy, and weighted cumulative residual entropy.

1. Terminological scope and principal formulations

The literature supports several recurrent WEGF forms, each adapted to a different information-theoretic or reliability-theoretic setting.

Form	Formula	Generated quantity
Direct weighted entropy generating function	$B_s(W,X)=\int_0^\infty x f^s(x)\,dx$	$B_s'(W,X)\big|_{s=1}=-H^w(X)$
General weighted information generating function	$f$0	$f$1
Rényi-type log-generator	$f$2	$f$3

In the direct lifetime-based definition, the weight is the outcome itself, $f$4, so larger lifetimes receive larger emphasis (S. et al., 20 Jul 2025). In the more general formulation, the weight becomes an arbitrary non-negative utility function $f$5, which recovers the direct definition when $f$6 (Saha et al., 2023). In the weighted Rényi setting, the generating object is typically the weighted $f$7-type integral or its logarithm; the weighted entropy is then a simple transform of that integral (Sekeh, 2015).

An abstract formulation is also available at the measure-decomposition level. Under the Condition of General Entropy Function (CGEF), weighted entropy can be written as

$f$8

where the masses $f$9 play the role of weights. This framework covers Shannon, Rényi, and Tsallis entropies, and the weighted and classical cover-based definitions coincide (Śmieja, 2013). A related weighted Rényi approach defines

$$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$0

again making the generating mechanism explicit through the $$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$1-power sum of decomposition weights (Śmieja et al., 2012).

A common misconception is that WEGF has a single canonical definition. The published record instead shows a family of closely related constructions: density-based, survival-based, cumulative-residual, relative, and measure-decomposition forms all obey the same generating logic, but they are tailored to different analytical problems (S. et al., 20 Jul 2025, S. et al., 2024, Saha et al., 2023).

2. Direct density-based WEGF

For a non-negative absolutely continuous random variable $$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$2 with pdf $$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$3, the explicit WEGF introduced in the reliability literature is

$$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$4

Equivalently,

$$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$5

Its central property is the generating relation

$$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$6

so that

$$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$7

with

$$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$8

Thus WEGF generalizes weighted entropy exactly as Golomb’s information generating function generalizes Shannon entropy (S. et al., 20 Jul 2025).

For standard models, $$B_s(W,X)=\int_0^\infty x f^s(x)\,dx,\qquad s\ge 0,\ s\neq 1,$$9 is often available in closed form:

Distribution	$s=1$0
Exponential $s=1$1	$s=1$2
Uniform on $s=1$3	$s=1$4
Lomax	$s=1$5
Power	$s=1$6
Pareto	$s=1$7

The direct WEGF is affine-covariant in a simple way. If $s=1$8 with $s=1$9, $H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$0, then

$H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$1

where $H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$2 is the unweighted entropy generating function (S. et al., 20 Jul 2025). This identity makes clear that weighted generation depends jointly on the scale and the shift, unlike the purely unweighted case.

The function also satisfies the lower bound

$H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$3

linking it to Shannon entropy and $H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$4 (S. et al., 20 Jul 2025). At the same time, a fixed value of $H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$5 does not determine the distribution. At $H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$6, $H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$7 and $H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$8 both satisfy

$H^w(X)=-\int_0^\infty x f(x)\log f(x)\,dx$9

so WEGF at a single parameter value is non-identifying (S. et al., 20 Jul 2025).

3. Residual and cumulative-residual variants

The dynamic extension of the direct WEGF is the weighted residual entropy generating function (WREGF),

$\omega$0

It satisfies $\omega$1 and the decomposition

$\omega$2

Differentiation yields the first-order linear ODE

$\omega$3

where $\omega$4 is the hazard rate (S. et al., 20 Jul 2025). This identity is structurally fundamental: it connects the generating function directly to the hazard process.

Several characterization results follow. If $\omega$5 is constant in $\omega$6 for some $\omega$7, then the hazard has power-law form and the distribution is Weibull. For $\omega$8, constant WREGF yields $\omega$9, hence a Pareto type I distribution (S. et al., 20 Jul 2025). The dynamic function also determines the underlying distribution uniquely, under the monotonicity condition stated in the uniqueness theorem. Two new classes of life distributions are then defined by monotonicity of $$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$0: increasing WREGF (IWREGF) and decreasing WREGF (DWREGF). Under these classes,

$$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$1

and, using

$$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$2

also

$$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$3

where $$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$4 is the mean residual life (S. et al., 20 Jul 2025).

A parallel construction replaces powers of the density by powers of the survival function. The weighted cumulative residual entropy generating function (WCREGF) is

$$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$5

and its dynamic version is

$$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$6

In that literature, $$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$7 denotes the survival function rather than the cdf. The derivative at $$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$8 generates weighted cumulative residual entropy, and the dynamic function obeys

$$I_\beta^\omega(X)=\int_0^\infty \omega(u)f^\beta(u)\,du,$$9

Consequently, the dynamic WCREGF uniquely determines the distribution. A further characterization states that the dynamic WCREGF is independent of $G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$0 if and only if the distribution is Rayleigh (S. et al., 2024).

These residual and cumulative-residual forms show that WEGF is not restricted to one-shot weighted entropy. It also provides a dynamic calculus for residual-life analysis, hazard-rate recovery, and lifetime characterization (S. et al., 20 Jul 2025, S. et al., 2024).

4. General weighted information generating functions

A broader and more flexible formulation is the general weighted information generating function (GWIGF),

$G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$1

with discrete counterpart

$G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$2

This construction subsumes the direct WEGF by taking $G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$3 (Saha et al., 2023).

Its derivatives are

$G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$4

Hence

$G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$5

At $G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$6, the same object yields weighted informational energy and weighted extropy (Saha et al., 2023).

The relative version,

$G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$7

plays the same generating role for weighted relative entropy. In particular,

$G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$8

the weighted Kullback–Leibler divergence (Saha et al., 2023).

Several structural properties distinguish the weighted setting. The function $G_\varphi(p;f)=\log\int \varphi(x)f(x)^p\,dx$9 is convex in $B_s(W,X)=\int_0^\infty x f^s(x)\,dx$0, but it is generally shift-dependent. For $B_s(W,X)=\int_0^\infty x f^s(x)\,dx$1 with $B_s(W,X)=\int_0^\infty x f^s(x)\,dx$2,

$B_s(W,X)=\int_0^\infty x f^s(x)\,dx$3

and, for the special weight $B_s(W,X)=\int_0^\infty x f^s(x)\,dx$4,

$B_s(W,X)=\int_0^\infty x f^s(x)\,dx$5

The theory also supplies bounds, transformation rules under monotone maps, ordering results via dispersive order, an upper bound for $B_s(W,X)=\int_0^\infty x f^s(x)\,dx$6 when $B_s(W,X)=\int_0^\infty x f^s(x)\,dx$7 and $B_s(W,X)=\int_0^\infty x f^s(x)\,dx$8 are independent, and explicit formulas for escort, generalized escort, and mixture distributions (Saha et al., 2023).

Residual variants are defined as well: $B_s(W,X)=\int_0^\infty x f^s(x)\,dx$9 with

$B_s'(W,X)\big|_{s=1}=-H^w(X)$0

so the generating principle persists under truncation and conditioning on survival (Saha et al., 2023).

5. Rényi, Tsallis, and abstract generating schemes

In weighted Rényi theory, the fundamental object is

$B_s'(W,X)\big|_{s=1}=-H^w(X)$1

A natural WEGF in this setting is

$B_s'(W,X)\big|_{s=1}=-H^w(X)$2

so that

$B_s'(W,X)\big|_{s=1}=-H^w(X)$3

The derivative at $B_s'(W,X)\big|_{s=1}=-H^w(X)$4 generates weighted Shannon entropy up to normalization: $B_s'(W,X)\big|_{s=1}=-H^w(X)$5 This interpretation is explicit in the analysis of maximum weighted Rényi entropy, where maximizing $B_s'(W,X)\big|_{s=1}=-H^w(X)$6 is equivalent to maximizing $B_s'(W,X)\big|_{s=1}=-H^w(X)$7 for fixed $B_s'(W,X)\big|_{s=1}=-H^w(X)$8 (Sekeh, 2015).

Under covariance constraints and compatibility conditions on the weight function, the maximizers are the Student-$B_s'(W,X)\big|_{s=1}=-H^w(X)$9 and Student-$f$00 families $f$01. The same framework yields explicit closed forms for the weighted Rényi entropies of these maximizers and an extended Hadamard inequality. In the broader interpretation supported there, the determinant inequality is a subadditivity statement for the joint versus marginal generating functions (Sekeh, 2015).

A measure-theoretic weighted Rényi approach reaches the same endpoint from a different direction. For a decomposition $f$02,

$f$03

and the corresponding weighted entropy of a cover equals the classical Rényi entropy of that cover. This equivalence is the central theorem of the weighted approach to Rényi entropy, and it is particularly useful for mixtures of measures and Rényi entropy dimensions (Śmieja et al., 2012).

An even more abstract generating scheme is provided by CGEF. If

$f$04

then the weighted version is

$f$05

Under CGEF, weighted and standard formulations coincide for Shannon, Rényi, and Tsallis entropies. This makes the pair $f$06 itself a generator of the entropy family, with the decomposition weights $f$07 providing the weighted state variables (Śmieja, 2013).

Taken together, these results show that WEGF may be interpreted narrowly as a single integral family such as $f$08, or more broadly as the generating layer underlying weighted Rényi, Tsallis, and general entropy constructions.

6. Structural inequalities, extremal principles, and rates

WEGF constructions are tightly linked to inequality theory. In the Gaussian setting, weighted entropy yields determinant inequalities through the formula

$f$09

where

$f$10

This identity underlies weighted Ky Fan, weighted Hadamard, weighted Szasz-type, weighted Toeplitz, and related determinant inequalities (Suhov et al., 2015).

The general weighted entropy literature develops weighted Gibbs inequalities, weighted Fano inequalities, weighted Ky Fan and Hadamard inequalities, and weighted Cramér–Rao inequalities through the weighted Fisher information matrix

$f$11

These results do not always name a WEGF explicitly, but they constrain the behavior of any parameterized weighted entropy generator through convexity, data-processing, and information-covariance relations (Suhov et al., 2015). An extended treatment also studies weighted entropy power

$f$12

weighted Lieb’s splitting inequality, and weighted Fisher information inequalities, providing further transform-based structures derived from weighted entropy (Kelbert et al., 2017).

The asymptotic theory of weighted entropy rates introduces a different generating viewpoint. For additive weight functions,

$f$13

the natural scale is $f$14, and the primary rate is

$f$15

under ergodicity and asymptotic additivity. For multiplicative weights,

$f$16

the natural scale is $f$17, and, in the Markov case,

$f$18

with $f$19 the leading eigenvalue of the weighted transfer operator $f$20 (Suhov et al., 2016). This suggests that WEGF is not only a finite-dimensional device; it also organizes entropy growth rates, spectral radii, and pressure-like limits.

7. Estimation, testing, and applications

WEGF-based constructions have moved from formal definition to statistical inference. For WREGF, a non-parametric goodness-of-fit test for Pareto type I distribution is built from the characterization that constant $f$21 corresponds to Pareto type I. The departure functional is

$f$22

with plug-in estimator

$f$23

The study includes extensive Monte Carlo simulation and two real-life datasets; for the Wheaton River flood exceedances data, the test fails to reject a Pareto type I model, whereas for the Rayleigh data it rejects the Pareto type I model at the $f$24 level (S. et al., 20 Jul 2025).

For dynamic WCREGF, the Rayleigh characterization produces a goodness-of-fit test based on the fact that constant dynamic WCREGF is equivalent to a Rayleigh distribution. The associated departure functional is

$f$25

estimated by a U-statistic. Monte Carlo experiments with $f$26 replicates assess empirical size and power against Weibull, Pareto, Lognormal, Half-normal, and linear failure rate alternatives. Real-data applications to ball bearing lifetimes and survival times of irradiated rats yield a non-rejection for the former and a rejection for the latter (S. et al., 2024).

The general weighted information generating function framework also supports estimation. A kernel-based estimator for the residual GWIGF is proposed,

$f$27

and its behavior is compared with a parametric plug-in estimator under exponential and fitted parametric models. The reported comparison is made in terms of absolute bias and mean squared error, with real-data illustrations from bladder cancer remission times and analgesic relief times (Saha et al., 2023).

These inferential developments show that WEGF is not merely a formal generator. It also supplies characterization identities, ordering criteria, non-parametric estimators, and model-assessment statistics in reliability and survival analysis.