Weighted Residual Entropy Generating Function

• Weighted Residual Entropy Generating Function (WREGF) is a residual-life extension of a weighted information generating function for continuous random variables, using conditional densities and survival functions.
• It establishes connections between weighted Shannon entropy, hazard rates, and mean residual life through differential identities, aiding in the characterization of various life distributions.
• WREGF serves as a foundation for both parametric and nonparametric estimation methods, offering practical tools for reliability testing and survival analysis.

Weighted Residual Entropy Generating Function (WREGF) is the residual-life version of a weighted information or entropy generating function for a nonnegative absolutely continuous random variable. In the formulation developed from the General Weighted Information Generating Function (GWIGF), if $X$ has density $f(x)$, survival function $\bar F(x)=P\{X>x\}$, inspection time $t\ge 0$, residual lifetime $X_t=[X-t\mid X>t]$, and nonnegative weight function $\omega(x)\ge 0$, then for order $\beta\ge 1$,

$$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$

In later work on the weighted entropy generating function, the specific weight $w(x)=x$ is used and the WREGF is written

$$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$

At $f(x)$0, both forms reduce to their corresponding static generating functions. Across these formulations, WREGF is studied as an age-dependent functional linked to weighted Shannon entropy, hazard rate, mean residual life, characterization of life distributions, and nonparametric estimation (Saha et al., 2023, S. et al., 20 Jul 2025).

1. Formal definition and residual-life construction

The residual formulation starts from the conditional density of the residual lifetime,

$f(x)$1

and inserts $f(x)$2 into the original weighted generating function. In the general weighted setting,

$f(x)$3

so the residual version is obtained by replacing $f(x)$4 with $f(x)$5 and integrating from $f(x)$6 to $f(x)$7: $f(x)$8 By construction, $f(x)$9 (Saha et al., 2023).

The later WREGF formulation specializes the weight to $\bar F(x)=P\{X>x\}$0: $\bar F(x)=P\{X>x\}$1 with static counterpart

$\bar F(x)=P\{X>x\}$2

This specialization preserves the residual-life interpretation while emphasizing larger lifetimes through the factor $\bar F(x)=P\{X>x\}$3 (S. et al., 20 Jul 2025).

A basic structural point is that the original GWIGF is shift-dependent, and the residual form inherits this feature. The residual quantity is therefore obtained by conditioning on survival past $\bar F(x)=P\{X>x\}$4, not by a mere shift in argument (Saha et al., 2023).

2. Entropic interpretation and generated quantities

A central property of the residual GWIGF is its connection to weighted Shannon entropy. Differentiating $\bar F(x)=P\{X>x\}$5 with respect to $\bar F(x)=P\{X>x\}$6 and evaluating at $\bar F(x)=P\{X>x\}$7 yields the negative of the residual weighted Shannon entropy: $\bar F(x)=P\{X>x\}$8 and

$\bar F(x)=P\{X>x\}$9

Thus the residual generating function acts as a generator for weighted residual entropy (Saha et al., 2023).

Special values recover additional information measures. In the general weighted setting,

$t\ge 0$0

where

$t\ge 0$1

is the residual weighted extropy. The same generating-function logic also motivates a residual weighted varentropy through second derivatives, by analogy with the unconditional relation

$t\ge 0$2

The residual version is obtained by differentiating the residual generating function for the conditional distribution (Saha et al., 2023).

For the $t\ge 0$3-weighted form, an alternative integral representation links WREGF to the ordinary residual entropy generating function

$t\ge 0$4

The identity is

$t\ge 0$5

This representation places WREGF within the broader family of residual generating functions rather than treating it as an isolated object (S. et al., 20 Jul 2025).

3. Hazard rate, mean residual life, and characterization theory

Differentiation with respect to the inspection time $t\ge 0$6 produces a direct link between WREGF and the hazard rate $t\ge 0$7. In the general weighted framework,

$t\ge 0$8

Hence, if $t\ge 0$9 is increasing in $X_t=[X-t\mid X>t]$0, then

$X_t=[X-t\mid X>t]$1

and if it is decreasing, the inequality reverses (Saha et al., 2023).

In the $X_t=[X-t\mid X>t]$2-weighted notation,

$X_t=[X-t\mid X>t]$3

or equivalently

$X_t=[X-t\mid X>t]$4

Using

$X_t=[X-t\mid X>t]$5

one obtains bounds in terms of mean residual life: $X_t=[X-t\mid X>t]$6 when $X_t=[X-t\mid X>t]$7 is increasing (resp. decreasing) in $X_t=[X-t\mid X>t]$8 (S. et al., 20 Jul 2025).

These differential identities support characterization results. If $X_t=[X-t\mid X>t]$9 is strictly increasing in $\omega(x)\ge 0$0, then the map $\omega(x)\ge 0$1 uniquely determines the distribution $\omega(x)\ge 0$2. Two constancy characterizations are especially explicit. For $\omega(x)\ge 0$3, $\omega(x)\ge 0$4 is constant in $\omega(x)\ge 0$5 if and only if $\omega(x)\ge 0$6 is Weibull, with failure rate

$\omega(x)\ge 0$7

For $\omega(x)\ge 0$8, $\omega(x)\ge 0$9 is constant in $\beta\ge 1$0 if and only if $\beta\ge 1$1 is Pareto (Type I), corresponding to $\beta\ge 1$2 (S. et al., 20 Jul 2025).

A comparison result based on hazard-rate ordering is available in the general weighted setting. If $\beta\ge 1$3 and either $\beta\ge 1$4 or $\beta\ge 1$5 has a decreasing failure-rate (DFR), then for every $\beta\ge 1$6 and any decreasing weight $\beta\ge 1$7,

$\beta\ge 1$8

This places WREGF among residual-life functionals that preserve ordering information derived from failure-rate structure (Saha et al., 2023).

4. Structural properties, transformations, and induced classes

Several identities describe how WREGF behaves under truncation and transformation. For the $\beta\ge 1$9-weighted form, the decomposition property is

$$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$0

This splits the static quantity into the contribution before time $$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$1 and the residual contribution after $$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$2 (S. et al., 20 Jul 2025).

Under a linear transformation $$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$3 with $$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$4 and $$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$5,

$$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$6

The corresponding transformed-formula viewpoint also appears in the broader GWIGF program, where generating functions of transformed random variables are obtained in terms of the generating function of a known distribution (Saha et al., 2023, S. et al., 20 Jul 2025).

Later work formalizes order and monotonicity classes generated by WREGF. One defines

$$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$7

and says that $$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$8 has Increasing Weighted Residual EGF (IWREGF) or Decreasing Weighted Residual EGF (DWREGF) according as $$I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.$$9 is increasing or decreasing. These are presented as two new classes of life distributions derived from WREGF (S. et al., 20 Jul 2025).

The exponential model furnishes a basic example. For $w(x)=x$0,

$w(x)=x$1

and the function is increasing in $w(x)=x$2, so the exponential law belongs to IWREGF (S. et al., 20 Jul 2025).

5. Closed forms, equilibrium models, and other explicit results

Closed-form WREGFs are available for standard lifetime models. For $w(x)=x$3 with density $w(x)=x$4, $w(x)=x$5, and weight $w(x)=x$6,

$w(x)=x$7

For $w(x)=x$8, $w(x)=x$9, $$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$0, with $$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$1,

$$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$2

These examples show that WREGF can be computed explicitly for both heavy-tailed and light-tailed models (Saha et al., 2023).

An equilibrium-distribution identity further connects WREGF to mean residual life. If $$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$3 is the equilibrium distribution of $$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$4, with density $$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$5 and $$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$6, then

$$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$7

where $$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$8 is the mean residual life of $$B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx, \qquad s\ge 0,\ s\neq 1.$$9 and $f(x)$00 is the weighted MRL of the proportional-hazards transform $f(x)$01 (Saha et al., 2023).

The sum of independent residual lifetimes also admits an upper-bound principle. For independent $f(x)$02 and $f(x)$03, the unconditional result

$f(x)$04

extends analogously when $f(x)$05 and $f(x)$06, yielding a corresponding inequality for the residual version $f(x)$07 (Saha et al., 2023).

Escort-distribution constructions remain part of the broader framework. The residual section does not work out the escort case explicitly, but Theorem 4.1 for the unconditional escort-GWIGF together with conditioning can be combined to obtain closed-form WREGFs for escort lifetimes (Saha et al., 2023).

6. Estimation, testing, applications, and related residual generating functions

WREGF has been studied with both nonparametric and parametric procedures. In the residual GWIGF treatment, given an i.i.d. sample $f(x)$08, a kernel density estimator

$f(x)$09

leads to

$f(x)$10

Simulation studies based on synthetic and real data show that bias and MSE of $f(x)$11 decrease as $f(x)$12 increases, and that both bias and MSE tend to grow with $f(x)$13 and with inspection time $f(x)$14 (Saha et al., 2023).

In the later WREGF treatment, the estimator is

$f(x)$15

Under standard kernel-density consistency conditions,

$f(x)$16

and asymptotic normality is derived via the delta method (S. et al., 20 Jul 2025).

A parametric comparison is available when $f(x)$17. Plugging the MLE $f(x)$18 into the known closed-form expression yields a parametric estimator that typically exhibits smaller bias and MSE than the nonparametric kernel version (Saha et al., 2023).

Testing methodology has also been developed from WREGF-based characterization. For Pareto Type I, the null hypothesis $f(x)$19 is assessed through

$f(x)$20

which is $f(x)$21 if $f(x)$22 is Pareto and $f(x)$23 otherwise. Its sample version,

$f(x)$24

uses order statistics and a kernel density estimate, and the null distribution is obtained by a parametric bootstrap under the estimated $f(x)$25. A Monte Carlo simulation study with $f(x)$26 replications over sample sizes $f(x)$27 reports empirical size close to the nominal $f(x)$28, and under alternatives including Gamma, Beta-Exponential, Inverse Beta, Benini, Weibull, Log-Normal, Half-Normal, and Tilted Pareto, the $f(x)$29-test often outperforms KS, Cramér–von Mises, Anderson–Darling, Zhang’s and Meintanis’s tests (S. et al., 20 Jul 2025).

Empirical illustrations span several survival-type datasets. For bladder-cancer remission times ($f(x)$30), goodness-of-fit tests $f(x)$31 identified the Generalized X-Exponential (GXE) model as best, and nonparametric WREGF estimates were computed at several $f(x)$32 and $f(x)$33. For analgesic relief times ($f(x)$34), the Gumbel type-II model fitted best, and nonparametric estimates were compared to parametric results from the assumed Gumbel fit (Saha et al., 2023). In the Pareto-testing study, flood-exceedance data from the Wheaton River ($f(x)$35) failed to reject the Pareto model, while Rayleigh-assumed lifetimes ($f(x)$36) led to rejection of the Pareto Type I hypothesis at the $f(x)$37 level (S. et al., 20 Jul 2025).

A related but distinct framework is the Weighted Cumulative Residual Entropy Generating Function (WCREGF),

$f(x)$38

which replaces the density power in WREGF by a survival-function power. Its dynamic version uniquely determines the distribution and is constant in $f(x)$39 if and only if $f(x)$40 is Rayleigh. This places WREGF within a wider family of residual generating functions indexed either by $f(x)$41 or by $f(x)$42 (S. et al., 2024).