Generalized Extropy Divergence Ratio (GEDR)

• GEDR is a normalized, directional measure that compares probability distributions via generalized extropy inaccuracy and self-extropy definitions.
• It accommodates density, CDF, and survival function representations, facilitating applications in lifetime analysis and image processing.
• The symmetric companion, GESR, quantifies similarity using a cosine-squared formulation, ensuring values are bounded between 0 and 1.

Searching arXiv for papers on extropy, GEDR, and related divergence ratios. arXiv search query: "Generalized Extropy Divergence Ratio extropy divergence ratio" Generalized Extropy Divergence Ratio (GEDR) is a class of extropy-based relative information measures for comparing two probability distributions through normalized cross-extropy. In the formulation introduced in "Extropy-Based Generalized Divergence and Similarity Ratios" (P. et al., 19 Aug 2025), GEDR quantifies directional discrepancy by dividing generalized extropy inaccuracy by generalized extropy itself, yielding a dimensionless ratio defined for density, distribution, and survival representations. Its construction belongs to the broader extropy program initiated by extropy as the complementary dual of entropy (Lad et al., 2011), but it differs from earlier extropy functionals in a crucial respect: it is explicitly comparative, directional, and normalized, whereas several antecedent measures in the literature were either single-distribution uncertainty indices or unnormalized divergence functionals.

1. Extropy foundations and the emergence of ratio-based comparison

Extropy was introduced as the complementary dual of entropy. For a finite discrete distribution $p_N=(p_1,\dots,p_N)$, entropy is

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$

whereas extropy is

$J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$

These coincide in the binary case, but for $N\ge 3$ they bifurcate into distinct measures; extropy is permutation-invariant, maximized at the uniform distribution, and its maximum remains bounded as $N\to\infty$ (Lad et al., 2011). In the continuous setting, the density analogue of extropy is

$j(f)=-\frac{1}{2}\int f^2(x)\,dx,$

which places extropy in an $L_2$-type geometry rather than the logarithmic geometry of entropy (Lad et al., 2011).

A central structural observation in the later GEDR framework is that many divergences can be written as the difference between an inaccuracy term and an average uncertainty term. Extropy naturally supplies such terms through inner products of probabilistic representations. Motivated by normalization for interpretability, in a manner analogous to correlation relative to covariance, GEDR was proposed as a ratio version of extropy-based comparison (P. et al., 19 Aug 2025).

The distinction from earlier extropy generalizations is important. "On Tsallis extropy with an application to pattern recognition" introduces Tsallis extropy,

$$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$

as a dual uncertainty measure to Tsallis entropy for a single discrete random variable, with limits to Shannon extropy and several order and boundedness properties, but it does not define a two-distribution divergence or any ratio-based construct named GEDR (Balakrishnan et al., 2021). This suggests that GEDR should be understood not as a direct consequence of Tsallis extropy, but as a later normalization program built on extropy-based inaccuracy.

2. Formal definition and representational scope

GEDR is defined on general probabilistic representations $\phi_1(x)$ and $\phi_2(x)$, where $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$0 may denote a PDF $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$1, a CDF $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$2, or a survival function $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$3, for nonnegative, absolutely continuous random variables on $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$4. The generalized extropy and generalized extropy inaccuracy are

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$5

The Generalized Extropy Divergence Ratio of $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$6 with $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$7 is then

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$8

with reverse ratio

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$9

Its symmetric companion, the Generalized Extropy Similarity Ratio (GESR), is

$J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$0

These are the core definitions of the framework (P. et al., 19 Aug 2025).

For discrete distributions with pmfs $J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$1 and $J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$2 on a common support, the same construction becomes

$J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$3

and

$J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$4

An explicit example given in the literature uses $J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$5 and $J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$6, yielding $J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$7, $J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$8, $J(p_N)=-\sum_{i=1}^N (1-p_i)\log(1-p_i).$9, hence $N\ge 3$0, $N\ge 3$1, and $N\ge 3$2 (P. et al., 19 Aug 2025).

The three principal continuous specializations are as follows.

Representation	Self-term	GEDR notation
Density $N\ge 3$3	$N\ge 3$4	$N\ge 3$5
Survival $N\ge 3$6	$N\ge 3$7	$N\ge 3$8
CDF $N\ge 3$9	$N\to\infty$0	$N\to\infty$1

In each case the cross-term is obtained by replacing the square with the product of the two representations, and the associated similarity ratio is the squared normalized cross-term (P. et al., 19 Aug 2025).

The domain requires finite negative denominators $N\to\infty$2 and finite cross-terms. For proper PDFs, CDFs, and survival functions this is ensured under mild square-integrability assumptions; support mismatches contribute zero through the product $N\to\infty$3, and the ratio is undefined only when the denominator vanishes, which is excluded by assumption (P. et al., 19 Aug 2025).

3. Axioms, range, and geometric interpretation

GEDR is asymmetric in general: $N\to\infty$4 It is strictly positive whenever the defining integrals are finite and the denominator is nonzero. It satisfies the identity condition

$N\to\infty$5

and more generally $N\to\infty$6 when $N\to\infty$7. Its range is therefore $N\to\infty$8, not $N\to\infty$9 with zero at equality; equality of distributions corresponds to the neutral value $j(f)=-\frac{1}{2}\int f^2(x)\,dx,$0, because GEDR is a normalized ratio rather than a difference-type divergence (P. et al., 19 Aug 2025).

A useful order relation links the two directions: $j(f)=-\frac{1}{2}\int f^2(x)\,dx,$1 The ratio identity

$j(f)=-\frac{1}{2}\int f^2(x)\,dx,$2

shows that directional imbalance is determined by the relative magnitudes of the two self-extropy terms (P. et al., 19 Aug 2025).

GESR removes directionality and restores boundedness: $j(f)=-\frac{1}{2}\int f^2(x)\,dx,$3 with $j(f)=-\frac{1}{2}\int f^2(x)\,dx,$4 iff $j(f)=-\frac{1}{2}\int f^2(x)\,dx,$5. The framework further establishes

$j(f)=-\frac{1}{2}\int f^2(x)\,dx,$6

in $j(f)=-\frac{1}{2}\int f^2(x)\,dx,$7, where

$j(f)=-\frac{1}{2}\int f^2(x)\,dx,$8

This gives a direct geometric meaning to GESR: values near $j(f)=-\frac{1}{2}\int f^2(x)\,dx,$9 indicate near alignment, and values near $L_2$0 indicate orthogonality. GEDR inherits interpretability from the product relation $L_2$1 (P. et al., 19 Aug 2025).

The framework explicitly states what GEDR is not. GEDR and GESR do not define a metric, because GEDR is asymmetric and neither satisfies the triangle inequality. They are not presented as $L_2$2-divergences or Bregman divergences; rather, they arise from extropy-based inner products and ratios. By contrast, the foundational extropy paper identified relative extropy and half the $L_2$3 metric within a Bregman-divergence perspective (Lad et al., 2011). This distinction is conceptually important: GEDR is a normalized ratio built on the extropy geometry, not another Bregman divergence.

The same paper establishes scale invariance,

$L_2$4

and location invariance under common shift,

$L_2$5

for nonnegative random variables under appropriate support behavior. Continuity under $L_2$6 perturbations is also stated (P. et al., 19 Aug 2025).

4. Structural bounds under stochastic models

A major part of the GEDR theory concerns bounds under structured stochastic relationships. Under the proportional generalized extropy model,

$L_2$7

the reverse directional ratio satisfies

$L_2$8

Consequently,

$L_2$9

together with

$$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$0

This gives a direct parametric link between asymmetry and similarity (P. et al., 19 Aug 2025).

For the proportional hazards model (PHM), where

$$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$1

equivalently $$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$2, the density-based and survival-based specializations admit one-sided bounds. In the density case, if $$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$3,

$$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$4

and for $$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$5,

$$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$6

with the inequalities reversed for $$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$7. In the survival case, if $$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$8,

$$JS_{\alpha}(X)=\frac{1}{\alpha-1}\left(N-1-\sum_{i=1}^N (1-p_i)^{\alpha}\right),$$9

and for $\phi_1(x)$0,

$\phi_1(x)$1

again reversed for $\phi_1(x)$2 (P. et al., 19 Aug 2025).

For the proportional reversed hazards model (PRHM), where

$\phi_1(x)$3

equivalently $\phi_1(x)$4, the CDF-based specialization satisfies, for $\phi_1(x)$5,

$\phi_1(x)$6

and for $\phi_1(x)$7,

$\phi_1(x)$8

with reversal for $\phi_1(x)$9. The corresponding similarity ratios remain bounded by $\phi_2(x)$0, with equality only at $\phi_2(x)$1 (P. et al., 19 Aug 2025).

These results clarify how GEDR reacts to systematic stochastic ordering. A plausible implication is that the sign of deviation from $\phi_2(x)$2 can encode model-relative dominance: in PHM the survival-based direction with the higher hazard tends to produce a ratio on one side of $\phi_2(x)$3, while the reverse direction lies on the opposite side.

5. Estimation, simulation, and applied use

For density-based comparison, GEDR is estimated via kernel density estimators. With independent samples $\phi_2(x)$4 and $\phi_2(x)$5, Parzen estimators

$\phi_2(x)$6

lead to

$\phi_2(x)$7

and hence

$\phi_2(x)$8

For survival and cumulative extropy versions, empirical survival functions and empirical CDFs are computed on a pooled grid $\phi_2(x)$9, and the required integrals are approximated by Riemann sums. The paper recommends choosing the probabilistic representation according to the application, constructing a common grid, using bandwidths $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$00 with $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$01, applying boundary correction on $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$02, and ensuring nonzero denominators and numerical stability (P. et al., 19 Aug 2025).

Simulation studies are reported for the similarity ratios. For $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$03 comparing $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$04 and $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$05, with true value $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$06, the bias decreases from $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$07 at $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$08 to $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$09 at $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$10, and the MSE decreases from $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$11 to $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$12. For $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$13 comparing $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$14 and $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$15, with true value $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$16, the bias drops from $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$17 to $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$18, and the MSE from $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$19 to $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$20. For $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$21 comparing $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$22 and $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$23, with true value $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$24, the bias also decreases with sample size (P. et al., 19 Aug 2025). These results are reported as confirmation of consistency.

The applications emphasize bounded similarity, but they are directly relevant to GEDR because $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$25. In lifetime data analysis of the FD&C Red No. 40 mouse experiment, self-similarity is approximately $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$26 on the diagonal, and similarity with the control group declines as dose increases; for example,

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$27

In image analysis, uniform intensity scaling by $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$28 preserves similarity: $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$29 and similarly for $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$30 and $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$31. A classification strategy based on a black reference image $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$32 uses $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$33 and $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$34 as exposure-invariant signatures (P. et al., 19 Aug 2025). Since GEDR is the directional factorization of the same similarity, these applications also illustrate the operational context in which directional discrepancy can be extracted when symmetry is not desired.

6. Relation to neighboring extropy measures and recurrent confusions

GEDR belongs to a family of extropy-based comparison measures, but the surrounding terminology is heterogeneous. A persistent source of confusion is the word “divergence.” In the Tsallis extropy literature, $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$35 is repeatedly called a “measure of discrimination,” yet it is a single-distribution uncertainty measure rather than a divergence between two distributions; the paper does not define a $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$36-versus-$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$37 construct and does not introduce any ratio named GEDR (Balakrishnan et al., 2021). Likewise, "Further results on relative, divergence measures based on extropy and their applications" develops extropy, relative extropy, extropy inaccuracy, extropy divergence, and dynamic residual and past variants, but explicitly does not define or use the term GEDR (P. et al., 10 Mar 2025).

A second distinction concerns unnormalized difference-type divergences. The foundational extropy work defines discrete relative extropy

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$38

and for densities identifies the extropic dual to KL divergence with half the $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$39 metric,

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$40

These are directed or symmetric discrepancy functionals, but they are not normalized ratios (Lad et al., 2011). Similarly, the survival-extropy divergence framework based on

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$41

supports U-statistic, EDF, and kernel estimators, as well as jackknife empirical likelihood inference, yet the paper states that it does not use the term GEDR; its main object is a squared $H(p_N)=-\sum_{i=1}^N p_i\log p_i,$42 survival-function divergence rather than a ratio (Garg et al., 21 Jul 2025).

A third distinction is with precursor ratio constructions. "Inaccuracy and divergence measures based on survival extropy" introduces the survival extropy inaccuracy ratio

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$43

together with asymmetric and symmetric survival-extropy divergences and dynamic versions, but not the generalized name GEDR (P. et al., 2024). This suggests that the survival-based specialization of GEDR,

$H(p_N)=-\sum_{i=1}^N p_i\log p_i,$44

can be viewed as a direct generalization and unification of such earlier ratio constructs, now placed within a common PDF/CDF/SF framework (P. et al., 19 Aug 2025).

Within this landscape, GEDR is best characterized as a normalized, directional extropy comparison functional with three distinguishing features: it is ratio-based rather than difference-based; it accommodates density, survival, and cumulative representations in a single formalism; and it is paired with a symmetric cosine-square similarity ratio. Those features mark the point at which extropy-based comparison shifts from isolated divergences and inaccuracies to a consolidated theory of relative information ratios.