Relative Cumulative Moment-Like Measures

• Relative cumulative moment-like measures are functionals that compare distributions via cumulative objects such as CDFs, survival functions, or Lorenz ordinates.
• They integrate cumulative quantities, applying weights or power transformations to capture information beyond pointwise density evaluations.
• These measures extend to frameworks in reliability theory, robust statistics, and information theory, underpinning generating functions and sharp inequalities.

Relative cumulative moment-like measures are functionals that compare distributions through cumulative objects—distribution functions, survival functions, Lorenz-curve ordinates, transformed cumulative coordinates, or representing measures—rather than through densities alone. In recent work, the term covers several technically distinct constructions: cdf-based quantities such as the cumulative past information generating function and its relative version, survival-based functionals such as the relative cumulative residual information, and biparametric relative cumulative moments induced by the relative differential-escort transform. Related frameworks extend the same viewpoint to Lorenz-type asymmetry measures, tail-integral moment representations, and measure-theoretic reweightings of moment sequences (Chaudhary et al., 2024, Andrews et al., 2024, Iagar et al., 23 Jul 2025, Schlemmer, 2022, Vila et al., 12 Jun 2026, Choi et al., 2018).

1. Conceptual scope and defining features

The common structural feature of these measures is that they integrate a cumulative quantity before or while taking powers, weights, or comparisons. For an absolutely continuous random variable $X$ with cdf $F$, the cumulative past information generating function is

$$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$

and for two random variables $X,Y$ with cdfs $F_X,F_Y$, the relative cumulative past information generating function is

$$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$

For non-negative lifetimes with survival functions $\bar F,\bar G$, the relative cumulative residual information is

$$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$

For two densities $f,h$ on a common support, the escort-based relative cumulative moment is

$$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$

with associated deviation-like quantity

$F$0

These constructions are cumulative because they are built from $F$1, $F$2, or a cumulative coordinate $F$3, and moment-like because they aggregate powers or weighted averages of such quantities (Chaudhary et al., 2024, Andrews et al., 2024, Iagar et al., 23 Jul 2025).

A second unifying feature is relativity. In some cases the comparison is direct, as in $F$4 or $F$5. In others, it is induced by a reference density or reference measure, as in $F$6 or in constructions where a representing measure $F$7 is replaced by $F$8. A third feature is that not every relative cumulative moment-like quantity is a divergence in the strict sense. Some are generating functions, some are overlap-type integrals, and some are bounded normalized functionals. This distinction is explicit in the literature: the CPIG-based divergence $F$9 is nonnegative, whereas the escort-based relative cumulative moment is not minimized at $$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$0, and the relative cumulative residual information is introduced as an information/divergence measure but not as a metric (Chaudhary et al., 2024, Iagar et al., 23 Jul 2025, Andrews et al., 2024).

2. Past-distribution formulations: CPIG, RCPIG, and Jensen constructions

The cdf-based branch of the subject is organized around the cumulative past information generating function

$$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$1

obtained from the cumulative information generating function by setting the survival exponent to zero. Its relative version,

$$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$2

mixes the past distribution of $$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$3 through $$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$4 with the past mass of $$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$5 through $$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$6. The construction is explicitly analogous to density-based information generating functions, but transferred from $$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$7 to $$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$8. This gives $$\mathcal{C}_\theta(X)=\int_\ell^r [F(x)]^\theta\,dx,\qquad \theta>0,$$9 the status of a cumulative past generating object, and $X,Y$0 the status of a relative cumulative past information generating measure (Chaudhary et al., 2024).

Its moment-like character is sharpened by the link with generalized cumulative past entropy. If

$X,Y$1

then

$X,Y$2

and

$X,Y$3

The first derivative at $X,Y$4 recovers cumulative past entropy, and the higher derivatives recover the generalized cumulative past entropies. At $X,Y$5, one also has

$X,Y$6

so the same family interpolates toward cumulative past extropy. The paper further derives lower bounds connecting CPIG to Shannon entropy, including

$X,Y$7

The relative and Jensenized side of this framework is built by combining cdfs through generalized logarithms and mixtures. The CPIG-based divergence

$X,Y$8

is nonnegative for all $X,Y$9. If $F_X,F_Y$0, the Jensen-cumulative past information generating function is defined with a sign depending on whether $F_X,F_Y$1 or $F_X,F_Y$2, and satisfies

$F_X,F_Y$3

There are parallel Jensen constructions for the fractional cumulative past entropy and the cumulative past Taneja entropy, with nonnegativity established for $F_X,F_Y$4 and $F_X,F_Y$5. The same paper also introduces a CPIG stochastic order, proves that dispersive order implies CPIG-order, and shows convolution bounds such as

$F_X,F_Y$6

with reversal for $F_X,F_Y$7. In this literature, relative cumulative moment-like measures are therefore not isolated objects but part of a cdf-based calculus involving generating functions, divergences, Jensen-type decompositions, stochastic orderings, and convolution inequalities (Chaudhary et al., 2024).

3. Survival-function formulations: RCRI and dynamic residual versions

A second major line of development uses survival functions rather than cdfs. For non-negative random variables $F_X,F_Y$8 with survival functions $F_X,F_Y$9, the relative cumulative residual information is

$$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$0

When $$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$1, this reduces to

$$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$2

which is exactly the cumulative residual entropy generating function. The dynamic version is defined through residual lifetimes $$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$3 and $$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$4, giving

$$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$5

At $$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$6, it recovers the static quantity. The measure is cumulative because it integrates powers of survival probabilities over time, and moment-like because $$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$7 is the classical mean identity from which these power-based generalizations depart (Andrews et al., 2024).

Under proportional hazards,

$$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$8

the integral collapses to

$$\mathcal{C}_\theta(X,Y)=\int_{-\infty}^{\infty}[F_X(x)]^\theta F_Y(x)\,dx,\qquad \theta>0.$$9

and the dynamic version satisfies an ODE involving the hazard rate: $\bar F,\bar G$0 In the general two-distribution case,

$\bar F,\bar G$1

These identities are the basis of several characterization theorems. If $\bar F,\bar G$2 is constant in $\bar F,\bar G$3, then $\bar F,\bar G$4 is exponential if and only if $\bar F,\bar G$5 is exponential. Under proportional hazards, linearity of $\bar F,\bar G$6 in $\bar F,\bar G$7 is equivalent to $\bar F,\bar G$8 being a Generalized Pareto Distribution with survival

$\bar F,\bar G$9

The same model is characterized by either

$$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$0

or

$$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$1

where $$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$2 is the mean residual life. Under affine scaling $$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$3 with $$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$4, $$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$5, and the proportional hazards model, the paper proves

$$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$6

so the measure scales linearly in time (Andrews et al., 2024).

The same work develops kernel-based nonparametric estimators. With

$$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$7

the plug-in estimator of $$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$8 is obtained by replacing $$R_{\alpha,\beta}(\bar F,\bar G)=\int_0^\infty \bar F^\alpha(x)\bar G^\beta(x)\,dx,\qquad \alpha,\beta>0.$$9 inside the defining integral, and similarly for $f,h$0 using conditional kernel survival estimates. Under $f,h$1 and $f,h$2, both estimators converge in probability to their population targets. The paper also reports an extensive Monte Carlo study and a Gaia DR3 application, where RCRI values between photometric bands of one object are close to one another, while a between-object comparison produces a markedly larger value (Andrews et al., 2024).

4. Escort-based relative cumulative moments and sharp information inequalities

A third, explicitly information-theoretic, formulation begins with two positive densities $f,h$3 on a common support and the relative differential-escort transform

$f,h$4

This transform turns the cumulative coordinate

$f,h$5

into the central object. The associated relative cumulative moment is then

$f,h$6

with

$f,h$7

When $f,h$8, the paper notes that

$f,h$9

so the relative family collapses to a non-relative cumulative moment built from the ordinary cdf. When $$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$0,

$$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$1

so the cumulative integral is purely that of the reference density $$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$2 (Iagar et al., 23 Jul 2025).

These relative cumulative moments are introduced together with a biparametric relative Fisher divergence

$$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$3

and the whole construction is organized around the fact that Shannon and Rényi quantities of the escort-transformed density become KL and Rényi divergences between $$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$4 and $$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$5. The basic identities are

$$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$6

and

$$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$7

Because of these correspondences, classical moment-entropy and Stam inequalities can be pulled back to the relative setting. One central bound is

$$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$8

and in the KL limit,

$$\mu_{p,\alpha}[f\mid h] = \int_{\Omega} \left| \int_{x_i}^x f(s)^{1-\alpha}h(s)^\alpha\,ds \right|^p f(x)\,dx,$$9

The paper emphasizes that these bounds are sharp and that the minimizers are obtained by applying the inverse relative differential-escort transformation to stretched Gaussian densities. It further gives explicit forms for the minimizing densities in terms of generalized trigonometric functions, generalized hyperbolic functions, and, in the $F$00 case, the inverse incomplete Gamma function (Iagar et al., 23 Jul 2025).

Several structural properties distinguish this family from other relative cumulative measures. First, it is scale-invariant: $F$01 matching the scaling behavior of KL and Rényi divergences under simultaneous dilation. Second, the paper explicitly notes that $F$02 is not minimized at $F$03; unlike a divergence, it is a relative cumulative moment rather than a discrepancy functional in the strict sense. Third, the same escort mechanism supports adapted inequalities for a prescribed minimizing density $F$04, where the reference $F$05 is constructed so that the product inequality is minimized only at $F$06. In that construction, generalized trigonometric functions again play a central role (Iagar et al., 23 Jul 2025).

5. Related constructions from Lorenz curves, tail integrals, moment sequences, and Gaussian benchmarks

The broader literature contains several constructions that, while not always presented under exactly the same terminology, fit the same cumulative-comparative template. In the Lorenz-curve setting, cumulative skewness is defined from ordered data by

$F$07

Here $F$08 is the cumulative proportion of observations, $F$09 is the cumulative proportion of the variable, and the weights are negative below the median rank and positive above it. The measure is bounded, with $F$10 and finite-sample bounds $F$11. The paper explicitly describes it as cumulative because it is built from Lorenz cumulative shares, and relative because it normalizes weighted asymmetry by total Lorenz deviation; it is also presented as robust to outliers compared with third-moment skewness (Schlemmer, 2022).

In the moment-sequence framework, relative cumulative constructions arise through transformations of representing measures. If $F$12 is an $F$13-moment sequence and $F$14 on $F$15, then

$F$16

is again an $F$17-moment sequence, with representing measure

$F$18

Subsequences $F$19 similarly correspond to multiplication by $F$20 and pushforward under $F$21. This suggests a measure-theoretic interpretation of relative cumulative moment-like measures in which a new moment functional is derived from a reference one by weighting and deterministic transformation, while existence, uniqueness, and support remain controlled by the Hausdorff or Stieltjes moment structure (Choi et al., 2018).

The real-order moment framework based on cdfs and tails supplies another closely related perspective. For arbitrary real-valued $F$22 and $F$23,

$F$24

and for $F$25, negative moments admit

$F$26

The paper does not explicitly define relative measures, but it states that the tail-integral structure naturally suggests definitions such as

$F$27

together with discrete analogues based on weighted tail sums and logarithmic analogues based on Laplace transforms. In this sense, cumulative moment-like comparison can be formulated directly as a weighted comparison of tails or cdfs (Vila et al., 12 Jun 2026).

A further related viewpoint compares cumulative moment profiles to a benchmark law. Under cumulant bounds of the form

$F$28

the deviation of a normalized variable $F$29 from the Gaussian benchmark $F$30 is quantified by

$F$31

The paper shows that $F$32 for even $F$33 and $F$34 for odd $F$35, and explicitly interprets cumulative sums of such deviations as cumulative moment-like measures of distance to Gaussianity. This is not a cumulative cdf- or survival-based construction, but it retains the same comparative aggregation logic (Eichelsbacher et al., 2019).

6. Comparative structure, misconceptions, and domains of use

Taken together, these literatures support a three-part description of relative cumulative moment-like measures. They are cumulative because they integrate $F$36, $F$37, Lorenz distances, cumulative coordinates, or weighted tails over the support. They are relative because they compare two distributions, one distribution against a reference density, one moment sequence against a reweighted version, or one moment profile against a benchmark law. They are moment-like because the defining objects are powers, weighted averages, or generating-function derivatives of cumulative quantities rather than pointwise density evaluations. This synthesis is consistent with the way the individual papers connect their measures to entropy, extropy, mean residual life, Hausdorff moments, or Gaussian moments (Chaudhary et al., 2024, Iagar et al., 23 Jul 2025, Andrews et al., 2024, Vila et al., 12 Jun 2026).

Several recurrent misconceptions are explicitly ruled out by the underlying results. First, relative cumulative moment-like measures are not uniformly divergences. The CPIG-based $F$38 is nonnegative, but $F$39 is not minimized at $F$40, and RCRI is introduced as an information/divergence measure without metric properties. Second, “relative” does not require logarithmic ratios; the comparison may be implemented by powers of cdfs or survival functions, by weighting a representing measure through $F$41, or by rank-weighted normalization in Lorenz space. Third, “cumulative” does not imply restriction to reliability theory. The same formal pattern appears in survival analysis, past-lifetime analysis, asymmetry measurement, moment problems, and information inequalities (Chaudhary et al., 2024, Andrews et al., 2024, Iagar et al., 23 Jul 2025, Schlemmer, 2022, Choi et al., 2018).

The principal application domains are correspondingly diverse. In reliability and lifetime analysis, CPIG, RCPIG, RCRI, and DRCRI quantify cumulative past or residual uncertainty and support characterization theorems under proportional hazards or generalized Pareto structure. In information theory, escort-based relative cumulative moments form one leg of a relative information triangle alongside scale-invariant relative Fisher measures and KL/Rényi divergences. In robust statistics, Lorenz-based cumulative skewness provides a bounded asymmetry functional that behaves similarly to third-moment skewness for “normal” distributions but is less sensitive to outliers. In applied data analysis, the Gaia DR3 study illustrates how relative cumulative residual information can compare photometric distributions within and across astronomical objects. In moment theory, weighted and pushed-forward representing measures furnish explicit relative constructions whose supports and determinacy properties can still be tracked (Andrews et al., 2024, Iagar et al., 23 Jul 2025, Schlemmer, 2022, Choi et al., 2018).

A plausible implication of this body of work is that the subject is best understood not as a single invariant, but as a family of comparison principles indexed by the cumulative object being integrated and by the manner in which reference information enters the construction. The mature part of the theory already includes exact definitions, sharp inequalities, stochastic orders, characterization results, and consistent nonparametric estimators; the less formal part, especially in tail-integral and moment-sequence settings, suggests additional relative cumulative measures that can be built whenever cumulative representations of moments or entropies are available (Vila et al., 12 Jun 2026, Chaudhary et al., 2024, Iagar et al., 23 Jul 2025).