arxiv: 2604.10555 · v1 · submitted 2026-04-12 · 📊 stat.OT

Recognition: unknown

On Some Multivariate Extensions to Zenga Curve: Properties and Applications

Shifna P R, S.M. Sunoj

Pith reviewed 2026-05-10 16:28 UTC · model grok-4.3

classification 📊 stat.OT

keywords Zenga curvebivariate inequalityquantile functionsmultidimensional inequalityZenga surfacesvector-valued curvedigital inequalitynonparametric estimation

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The pith

Bivariate Zenga surfaces defined from quantile functions extend inequality measurement to the joint distribution of two variables.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to address the shortcoming of one-dimensional inequality tools by constructing surfaces and curves that operate directly on pairs of variables. It does so through bivariate quantile functions that map probability levels to pairs of values, then adapts the Zenga construction to these functions. The resulting surfaces and vector-valued curves are meant to reveal how inequality in one variable changes across levels of the other. The authors supply theoretical properties, a practical estimator, and an application to broadband and digital-literacy data across countries. A reader who accepts the definitions will view them as a route to more faithful descriptions of linked disparities.

Core claim

Bivariate quantile functions yield well-defined Zenga surfaces whose values at each probability pair quantify inequality in the joint distribution, together with a vector-valued curve that decomposes the inequality contribution of each margin; these objects satisfy natural ordering and monotonicity properties and admit consistent nonparametric estimation.

What carries the argument

Bivariate quantile functions, which replace the univariate inverse distribution function and serve as the direct input for constructing the Zenga surfaces and the associated vector-valued curve.

If this is right

The surfaces make visible the conditional inequality of one variable given the quantile level of the other.
The vector-valued curve supplies separate components for each variable while preserving the joint information.
The nonparametric estimator can be applied directly to sample data without parametric assumptions on the joint law.
Real-world application to paired digital indicators demonstrates that the surfaces detect inequality patterns missed by separate univariate calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same quantile-function route could be iterated to three or more variables, producing hypersurfaces for broader multidimensional inequality studies.
Policy analysts could use the surfaces to locate regions of the joint distribution where two disadvantages reinforce each other.
Direct numerical comparison with other bivariate inequality functionals on the same datasets would clarify whether the Zenga construction adds distinct interpretive value.

Load-bearing premise

The quantile-function definitions produce numerical values that correspond to the intuitive notion of multidimensional inequality in the joint distribution.

What would settle it

A controlled simulation in which two variables are made increasingly dependent while each marginal inequality is held fixed, yet the proposed surface values remain unchanged or move in the opposite direction from the expected rise in joint disparity.

Figures

Figures reproduced from arXiv: 2604.10555 by Shifna P R, S.M. Sunoj.

**Figure 2.** Figure 2: Bivariate Zenga Surface for Power distribution Hence, the joint distribution function can be written as F(x1, x2) = x1 K1 1/b1 K1x2 K2x1 1/b2 . For 0 < x1, x2 < 1 this is a bivariate power distribution. Then the Zenga surface (3.5) becomes, I(u1, u2) = 1 − u b1+1 1 − u b2+1 2 (u b1+1 1 − 1)(u b2+1 2 − 1) . The plot for bivariate Zenga surface of the power distribution for b1 = 2, b2 = 3 is given in … view at source ↗

**Figure 3.** Figure 3: Graphical representation for I12(u1, u2) for Pareto distribution [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

**Figure 4.** Figure 4: Graphical representation for I21(u1, u2) for Pareto distribution 20 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗

**Figure 5.** Figure 5: Contour plot of the estimated bivariate Zenga surface ˆI12(u1, u2), representing inequality in broadband access conditional on digital skills [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗

**Figure 6.** Figure 6: Contour plot of the estimated bivariate Zenga surface ˆI21(u1, u2), representing inequality in digital skills conditional on broadband access [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗

**Figure 7.** Figure 7: Slice plots of ˆI12(u1, u2) for selected values of u2, illustrating the variation of broadband inequality across different quantile levels of digital skills. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗

**Figure 8.** Figure 8: Slice plots of ˆI21(u1, u2) for selected values of u1, showing the variation of digital skill inequality across different quantile levels of broadband access. broadband access improves. An important observation from both the contour and slice plots is the asymmetric nature of the two measures. The surfaces ˆI12(u1, u2) and ˆI21(u1, u2) exhibit different patterns, indicating that inequality in one dimension… view at source ↗

read the original abstract

Measures of inequality are often limited in their ability to capture multidimensional aspects that arise from the joint distribution of multiple socio-economic variables. In this paper, we develop bivariate extensions of the Zenga inequality measure using bivariate quantile functions. We propose new bivariate Zenga surfaces and study their theoretical properties. A vector-valued bivariate Zenga curve is also introduced to provide a more detailed characterization of inequality. A non-parametric estimator is proposed and methods are evaluated through simulation studies and applied to the analysis of digital inequality across countries using indicators such as broadband penetration and digital literacy. The results highlight the effectiveness of the proposed framework in capturing multidimensional inequality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a direct bivariate extension of the Zenga measure with new surfaces and a vector curve, but the non-unique bivariate quantile definition leaves the surfaces open to arbitrary choices.

read the letter

The main point is that this work defines bivariate Zenga surfaces and a vector-valued curve from bivariate quantile functions, adds a non-parametric estimator, runs simulations, and applies the tools to digital inequality data across countries using broadband and literacy indicators. It builds straightforwardly on the univariate Zenga idea to handle joint distributions. The simulations and country-level example are useful for seeing how the measures behave in practice and for spotting patterns that single-variable versions miss. The properties they derive look like routine extensions of the original definitions. The application is concrete and shows the framework can be computed on real socio-economic data. The soft spot is the quantile foundation. Bivariate quantiles have no canonical form, and choices such as conditional or spatial versions produce different surfaces. The paper adopts one without comparing alternatives or showing invariance, so the claim that the surfaces accurately reflect multidimensional inequality rests on that unexamined choice. The estimator is presented as non-parametric, but details on its behavior under dependence or sample size are thin in the abstract. This is aimed at statisticians and economists who already use Zenga-type indices and want a two-variable version for policy work. A reader tracking inequality measurement tools would find the constructions and the digital example worth a look. The thinking is clear and the literature engagement is direct. I would bring it to a reading group as maybe, would not cite it in the next year, and would send it for peer review so the quantile choice and estimator properties get checked.

Referee Report

2 major / 2 minor

Summary. The paper develops bivariate extensions of the Zenga inequality measure using bivariate quantile functions. It proposes new bivariate Zenga surfaces, studies their theoretical properties, introduces a vector-valued bivariate Zenga curve for more detailed characterization, proposes a non-parametric estimator, evaluates it via simulation studies, and applies the framework to digital inequality across countries using indicators such as broadband penetration and digital literacy.

Significance. If the central definitions are made robust and the properties hold under a canonical choice, the work could offer useful tools for multidimensional inequality analysis that capture joint distributional features beyond univariate Zenga measures. The combination of theoretical properties, simulations, and a real-data application on digital inequality adds practical value, though the overall significance hinges on resolving ambiguities in the bivariate quantile construction.

major comments (2)

[Abstract and §3] The central construction of the bivariate Zenga surfaces and vector-valued curve (abstract; likely §3) relies on bivariate quantile functions, but no unique canonical definition exists (unlike the univariate Q(p) = inf{x : F(x) ≥ p}). The manuscript adopts one specific version without proving invariance properties or comparing alternatives such as geometric, conditional, or spatial quantiles; this choice directly determines the level sets and thus the inequality surfaces, leaving the claim that they 'accurately reflect multidimensional inequality aspects in joint distributions' unsecured by the construction itself.
[§4 and application section] The theoretical properties studied for the proposed surfaces (likely §4) and the effectiveness highlighted in the application (§6 or §7) rest on the above definitions; without addressing non-uniqueness, the properties may not be intrinsic and the simulation results may not generalize across plausible bivariate quantile choices.

minor comments (2)

[Abstract] The abstract states that 'the results highlight the effectiveness' but does not include any quantitative summary of simulation performance or application findings; adding one or two key metrics would improve clarity.
[§3] Notation for the bivariate quantile function and the resulting surfaces should be introduced with explicit reference to the chosen definition to avoid ambiguity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments, which help clarify important aspects of our bivariate extensions. We address each major comment below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract and §3] The central construction of the bivariate Zenga surfaces and vector-valued curve (abstract; likely §3) relies on bivariate quantile functions, but no unique canonical definition exists (unlike the univariate Q(p) = inf{x : F(x) ≥ p}). The manuscript adopts one specific version without proving invariance properties or comparing alternatives such as geometric, conditional, or spatial quantiles; this choice directly determines the level sets and thus the inequality surfaces, leaving the claim that they 'accurately reflect multidimensional inequality aspects in joint distributions' unsecured by the construction itself.

Authors: We appreciate the referee highlighting the non-uniqueness of bivariate quantile functions. Our paper adopts the definition of the bivariate quantile function obtained as the generalized inverse of the joint cumulative distribution function, which provides a direct and natural multivariate extension of the univariate quantile used in the original Zenga measure. While the original manuscript did not include an explicit comparison to alternatives (such as conditional, geometric, or spatial quantiles), we agree this should be addressed for transparency. In the revised version, we will expand §3 to (i) state the chosen definition explicitly, (ii) briefly discuss why it is suitable for inequality analysis (preserving monotonicity and allowing level-set interpretations for joint distributions), and (iii) note that other definitions exist and may yield different surfaces. We will also tone down the abstract claim to indicate that the surfaces accurately reflect multidimensional aspects under this construction. These changes secure the claims without requiring invariance proofs across all definitions. revision: yes
Referee: [§4 and application section] The theoretical properties studied for the proposed surfaces (likely §4) and the effectiveness highlighted in the application (§6 or §7) rest on the above definitions; without addressing non-uniqueness, the properties may not be intrinsic and the simulation results may not generalize across plausible bivariate quantile choices.

Authors: We concur that the properties in §4 and the simulation/application results are specific to the adopted bivariate quantile definition. In the revision, we will modify §4 to state explicitly that all derived properties (e.g., bounds, monotonicity with respect to joint distributions) hold for the proposed surfaces under the chosen quantile construction. In the simulation studies, we will add a clarifying sentence noting that estimator performance is assessed for this definition and that results pertain to it. For the digital inequality application, we will similarly emphasize that the observed patterns illustrate the framework with the selected quantile. While a full comparative simulation across multiple quantile definitions lies outside the paper's scope, the revisions will make the dependence clear and prevent any implication that the properties are definition-independent. revision: partial

Circularity Check

0 steps flagged

No circularity: bivariate Zenga extensions are explicit new definitions

full rationale

The paper starts from the univariate Zenga measure and explicitly constructs bivariate extensions via bivariate quantile functions, then derives properties of the resulting surfaces and vector-valued curve, proposes a nonparametric estimator, and validates via simulation and data application. None of these steps reduce by construction to fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations; the central objects are introduced as novel constructs whose properties are examined independently. The provided abstract and description contain no equations or claims that collapse the output back to the input definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

From the abstract alone, the central claim rests on new definitions of bivariate surfaces and curves derived from quantile functions, with no explicit free parameters, axioms, or invented entities detailed. The work assumes standard properties of quantile functions carry over to the bivariate case.

pith-pipeline@v0.9.0 · 5395 in / 1131 out tokens · 44953 ms · 2026-05-10T16:28:55.707458+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references

[1]

and Rigo, P

Berti, P. and Rigo, P. (1995). A note on Zenga concentration index.Journal of the Italian Statistical Society, 4(3):397–404. Billon, M., Marco, R., and Lera-Lopez, F. (2009). Disparities in ICT adoption: A mul- tidimensional approach to study the cross-country digital divide.Telecommunications Policy, 33(10-11):596–610. 28 Chinn, M. D. and Fairlie, R. W. ...

1995
[2]

Greselin, F., Pasquazzi, L., and Zitikis, R. (2010). Zenga’s new index of economic in- equality, its estimation, and an analysis of incomes in italy.Journal of Probability and Statistics, 2010(1):718905. Jain, K. and Nanda, A. K. (1995). On multivariate weighted distributions.Communica- tions in Statistics-Theory and Methods, 24(10):2517–2539. Langel, M. ...

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[3]

29 Shifna, P. R. and Sunoj, S. M. (2025). Extending the Lorenz curve to higher dimensions: A multivariate quantile function approach.Research Square. Preprint. Thoene, U. and García Alonso, R. (2025). Broadband disparities and policy responses in Latin America and the Caribbean.Digital Policy, Regulation and Governance, 27(5):588–606. Vineshkumar, B. and ...

2025