---
title: Weighted Residual Entropy Generating Function
url: https://www.emergentmind.com/topics/weighted-residual-entropy-generating-function-wregf
type: topic
---

# Weighted Residual Entropy Generating Function

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 \(t=0\), 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 [2305.18746; 2507.15057].

## 1. Formal definition and residual-life construction

The residual formulation starts from the conditional density of the residual lifetime,
\[
f_t(x)=\frac{f(x)}{\bar F(t)},\qquad x>t,
\]
and inserts \(f_t\) into the original weighted generating function. In the general weighted setting,
\[
I_\beta^\omega(X)=\int_0^\infty \omega(x)\,f(x)^\beta\,dx,
\]
so the residual version is obtained by replacing \(f(x)\) with \(f_t(x)\) and integrating from \(t\) to \(\infty\):
\[
I^{\omega}_\beta(X;t)=\int_t^\infty \omega(x)\,\bigl(f_t(x)\bigr)^\beta\,dx
=\int_t^\infty \omega(x)\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^\beta\,dx.
\]
By construction, \(I_\beta^\omega(X;0)=I_\beta^\omega(X)\) [2305.18746].

The later WREGF formulation specializes the weight to \(x\):
\[
B_s(W,X;t)=\int_t^\infty x\,\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx,
\]
with static counterpart
\[
B_s(W,X)=\int_0^\infty x\,f^s(x)\,dx
=E\bigl[X\,f^{\,s-1}(X)\bigr].
\]
This specialization preserves the residual-life interpretation while emphasizing larger lifetimes through the factor \(x\) [2507.15057].

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 \(t\), not by a mere shift in argument [2305.18746].

## 2. Entropic interpretation and generated quantities

A central property of the residual GWIGF is its connection to weighted Shannon entropy. Differentiating \(I^\omega_\beta(X;t)\) with respect to \(\beta\) and evaluating at \(\beta=1\) yields the negative of the residual weighted Shannon entropy:
\[
H^{\omega}(X;t)
=-\int_t^\infty \omega(x)\,\frac{f(x)}{\bar F(t)}\,
\log\frac{f(x)}{\bar F(t)}\,dx,
\]
and
\[
\frac{\partial}{\partial\beta}I^\omega_\beta(X;t)\Big|_{\beta=1}
=-\,H^\omega(X;t).
\]
Thus the residual generating function acts as a generator for weighted residual entropy [2305.18746].

Special values recover additional information measures. In the general weighted setting,
\[
I^\omega_1(X;t)=E[\omega(X)\mid X>t],
\qquad
I^\omega_2(X;t)=-2\,J^\omega(X;t),
\]
where
\[
J^\omega(X;t)=-\tfrac12\int_t^\infty
\omega(x)\Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^2\,dx
\]
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
\[
\mathrm{VarEnt}^\omega(X)
=
\frac{\partial^2}{\partial\beta^2}I^{\omega^2}_\beta(X)\Big|_{\beta=1}
-
\Bigl[
\frac{\partial}{\partial\beta}I^\omega_\beta(X)\Big|_{\beta=1}
\Bigr]^2.
\]
The residual version is obtained by differentiating the residual generating function for the conditional distribution [2305.18746].

For the \(x\)-weighted form, an alternative integral representation links WREGF to the ordinary residual entropy generating function
\[
B_s(X;t)=\int_t^\infty \Bigl(\frac{f(x)}{\bar F(t)}\Bigr)^s\,dx.
\]
The identity is
\[
B_s(W,X;t)
=
t\,B_s(X;t)
+
\int_t^\infty
\Bigl(\frac{\bar F(x)}{\bar F(t)}\Bigr)^s
B_s(X;x)\,dx.
\]
This representation places WREGF within the broader family of residual generating functions rather than treating it as an isolated object [2507.15057].

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

Differentiation with respect to the inspection time \(t\) produces a direct link between WREGF and the hazard rate \(h(t)=f(t)/\bar F(t)\). In the general weighted framework,
\[
\frac{d}{dt}I^\omega_\beta(X;t)
=
-\,\omega(t)\,h(t)^\beta
+
\beta\,h(t)\,I^\omega_\beta(X;t).
\]
Hence, if \(I^\omega_\beta(X;t)\) is increasing in \(t\), then
\[
I^\omega_\beta(X;t)\ge \frac1\beta\,\omega(t)\,h(t)^{\beta-1},
\]
and if it is decreasing, the inequality reverses [2305.18746].

In the \(x\)-weighted notation,
\[
\frac{d}{dt}B_s(W,X;t)
=
s\,h(t)\,B_s(W,X;t)-t\,[h(t)]^s,
\]
or equivalently
\[
B_s'(W,X;t)-s\,h(t)\,B_s(W,X;t)
=
-\,t\,[h(t)]^s.
\]
Using
\[
h(t)=\frac{1+m'(t)}{m(t)},
\qquad
m(t)=E[X-t\mid X>t],
\]
one obtains bounds in terms of mean residual life:
\[
B_s(W,X;t)
\ge
\frac{t}{s}\,\bigl[h(t)\bigr]^{s-1}
=
\frac{t}{s}\,
\Bigl(\frac{1+m'(t)}{m(t)}\Bigr)^{s-1}
\quad
(\text{resp.\ }\le)
\]
when \(B_s(W,X;t)\) is increasing (resp. decreasing) in \(t\) [2507.15057].

These differential identities support characterization results. If \(B_s(W,X;t)\) is strictly increasing in \(t\), then the map \(t\mapsto B_s(W,X;t)\) uniquely determines the distribution \(F\). Two constancy characterizations are especially explicit. For \(s\neq 2\), \(B_s(W,X;t)\) is constant in \(t\) if and only if \(X\) is Weibull, with failure rate
\[
h(t)=\Bigl(\frac{s\,k}{t}\Bigr)^{1/(s-1)}.
\]
For \(s=2\), \(B_2(W,X;t)\) is constant in \(t\) if and only if \(X\) is Pareto (Type I), corresponding to \(h(t)\propto 1/t\) [2507.15057].

A comparison result based on hazard-rate ordering is available in the general weighted setting. If \(Y\le_{\rm hr}X\) and either \(X\) or \(Y\) has a decreasing failure-rate (DFR), then for every \(\beta\ge 1\) and any decreasing weight \(\omega(x)\),
\[
I^\omega_\beta(X;t)\le I^\omega_\beta(Y;t).
\]
This places WREGF among residual-life functionals that preserve ordering information derived from failure-rate structure [2305.18746].

## 4. Structural properties, transformations, and induced classes

Several identities describe how WREGF behaves under truncation and transformation. For the \(x\)-weighted form, the decomposition property is
\[
B_s(W,X)
=
\int_0^t x\,f^s(x)\,dx
+
(\bar F(t))^s\,B_s(W,X;t).
\]
This splits the static quantity into the contribution before time \(t\) and the residual contribution after \(t\) [2507.15057].

Under a linear transformation \(Y=aX+b\) with \(a>0\) and \(b\ge 0\),
\[
B_s\bigl(W,Y;t\bigr)
=
a^{1-s}
\Bigl[
a\,B_s\!\Bigl(W,X;\frac{t-b}{a}\Bigr)
+
b\,B_s\!\Bigl(X;\frac{t-b}{a}\Bigr)
\Bigr].
\]
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 [2305.18746; 2507.15057].

Later work formalizes order and monotonicity classes generated by WREGF. One defines
\[
X\le_{\rm WREGF}Y
\quad\text{if}\quad
B_s(W,X;t)\le B_s(W,Y;t)\ \ \forall\, t>0,
\]
and says that \(X\) has Increasing Weighted Residual EGF (IWREGF) or Decreasing Weighted Residual EGF (DWREGF) according as \(t\mapsto B_s(W,X;t)\) is increasing or decreasing. These are presented as two new classes of life distributions derived from WREGF [2507.15057].

The exponential model furnishes a basic example. For \(X\sim \mathrm{Exp}(\lambda)\),
\[
B_s(W,X;t)
=
\frac{\lambda^s\bigl(1+\lambda s\,t\bigr)}{(\lambda s)^2},
\]
and the function is increasing in \(t\), so the exponential law belongs to IWREGF [2507.15057].

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

Closed-form WREGFs are available for standard lifetime models. For \(X\sim\mathrm{ParetoI}(a,\gamma)\) with density \(f(x)=a\gamma^a/x^{a+1}\), \(x>\gamma\), and weight \(\omega(x)=x\),
\[
I_\beta^{x}(X;t)
=
\frac{1}{a-1}\,
\Bigl(\frac{a\,\gamma^a}{1-(\gamma/t)^a}\Bigr)^\beta
\,t^{\,1-a},
\qquad
a>1,\ \beta\ge 1.
\]
For \(X\sim\mathrm{Exp}(\lambda)\), \(f(x)=\lambda e^{-\lambda x}\), \(x>0\), with \(\omega(x)=x\),
\[
I_\beta^{x}(X;t)
=
\frac{\lambda^\beta}{(\beta\lambda)^2}\,
\bigl(\beta\lambda\,t+1\bigr),
\qquad \beta\ge 1.
\]
These examples show that WREGF can be computed explicitly for both heavy-tailed and light-tailed models [2305.18746].

An equilibrium-distribution identity further connects WREGF to mean residual life. If \(X_E\) is the equilibrium distribution of \(X\), with density \(f_E(x)=\bar F(x)/\mu\) and \(\mu=E(X)\), then
\[
I_\beta^\omega(X_E;t)
=
\frac{M_{X_\beta}^\omega(t)}{\bigl(M_X(t)\bigr)^\beta},
\]
where \(M_X(t)=E[X-t\mid X>t]\) is the mean residual life of \(X\) and \(M_{X_\beta}^\omega(t)\) is the weighted MRL of the proportional-hazards transform \(X_\beta\) [2305.18746].

The sum of independent residual lifetimes also admits an upper-bound principle. For independent \(X\) and \(Y\), the unconditional result
\[
I_\beta^x(X+Y)
\le
I_\beta(X)\,I_\beta^y(Y)
+
I_\beta^x(X)\,I_\beta(Y)
\]
extends analogously when \(X\mapsto X_t\) and \(Y\mapsto Y_t\), yielding a corresponding inequality for the residual version \(I_\beta^x(X+Y;t)\) [2305.18746].

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 [2305.18746].

## 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 \(X_1,\dots,X_n\), a kernel density estimator
\[
\widehat f(x)=\frac1{n\,b_n}\sum_{i=1}^n
k\!\Bigl(\frac{x-X_i}{b_n}\Bigr),
\qquad
b_n\to 0,\ \ n\,b_n\to\infty,
\]
leads to
\[
\widehat I_\beta^\omega(X;t)
=
\int_t^\infty
\omega(x)\,
\Bigl(\frac{\widehat f(x)}{\widehat{\bar F}(t)}\Bigr)^\beta dx,
\qquad
\widehat{\bar F}(t)=\int_t^\infty \widehat f(x)\,dx.
\]
Simulation studies based on synthetic and real data show that bias and MSE of \(\widehat I_\beta^\omega(X;t)\) decrease as \(n\) increases, and that both bias and MSE tend to grow with \(\beta\) and with inspection time \(t\) [2305.18746].

In the later WREGF treatment, the estimator is
\[
\hat f(x)
=
\frac1{n\,h}\sum_{j=1}^n
k\!\Bigl(\frac{x-X_j}{h}\Bigr),
\qquad
\widehat B_s(W,X;t)
=
\int_t^\infty
x\,
\Bigl(\frac{\hat f(x)}{\hat{\bar F}(t)}\Bigr)^s dx,
\qquad
\hat{\bar F}(t)=\frac1n\sum_{j=1}^n \mathbf1\{X_j>t\}.
\]
Under standard kernel-density consistency conditions,
\[
\widehat B_s(W,X;t)\xrightarrow{p}B_s(W,X;t),
\]
and asymptotic normality is derived via the delta method [2507.15057].

A parametric comparison is available when \(X\sim \mathrm{Exp}(\lambda)\). Plugging the MLE \(\widehat\lambda\) into the known closed-form expression yields a parametric estimator that typically exhibits smaller bias and MSE than the nonparametric kernel version [2305.18746].

Testing methodology has also been developed from WREGF-based characterization. For Pareto Type I, the null hypothesis \(H_0\!:F\in\mathrm{Pareto}(\alpha)\) is assessed through
\[
\Delta(F)
=
\int_1^\infty
\bigl[\,3x\,F(x)\,f^2(x)-x\,f^2(x)\bigr]\,dx,
\]
which is \(0\) if \(F\) is Pareto and \(>0\) otherwise. Its sample version,
\[
\widehat\Delta
=
\frac1{n^2}\sum_{i=1}^n
(3\,i-n)\,
X_{(i)}\,
\hat f\bigl(X_{(i)}\bigr),
\]
uses order statistics and a kernel density estimate, and the null distribution is obtained by a parametric bootstrap under the estimated \(\hat\alpha\). A Monte Carlo simulation study with \(10\,000\) replications over sample sizes \(n=10,25,50,75,100\) reports empirical size close to the nominal \(5\%\), and under alternatives including Gamma, Beta-Exponential, Inverse Beta, Benini, Weibull, Log-Normal, Half-Normal, and Tilted Pareto, the \(\Delta\)-test often outperforms KS, Cramér–von Mises, Anderson–Darling, Zhang’s and Meintanis’s tests [2507.15057].

Empirical illustrations span several survival-type datasets. For bladder-cancer remission times (\(n=128\)), goodness-of-fit tests \((-\log L,\ \mathrm{AIC},\ \mathrm{BIC})\) identified the Generalized X-Exponential (GXE) model as best, and nonparametric WREGF estimates were computed at several \(\beta\) and \(t\). For analgesic relief times (\(n=20\)), the Gumbel type-II model fitted best, and nonparametric estimates were compared to parametric results from the assumed Gumbel fit [2305.18746]. In the Pareto-testing study, flood-exceedance data from the Wheaton River (\(n=72\)) failed to reject the Pareto model, while Rayleigh-assumed lifetimes (\(n=14\)) led to rejection of the Pareto Type I hypothesis at the \(5\%\) level [2507.15057].

A related but distinct framework is the Weighted Cumulative Residual Entropy Generating Function (WCREGF),
\[
C_s^{(W)}(F;t)=\int_t^\infty x\,\Bigl[\frac{\bar F(x)}{\bar F(t)}\Bigr]^s\,dx,
\]
which replaces the density power in WREGF by a survival-function power. Its dynamic version uniquely determines the distribution and is constant in \(t\) if and only if \(X\) is Rayleigh. This places WREGF within a wider family of residual generating functions indexed either by \(f\) or by \(\bar F\) [2402.06571].

Source: https://www.emergentmind.com/topics/weighted-residual-entropy-generating-function-wregf