FRoM-W₁: Unified Quantile Measure Framework

• The framework is a unified approach that represents probability measures via quantile functions, encapsulating data necessary for computing Lorenz curves and inequality indices.
• It guarantees rigorous convergence under the Wasserstein-1 metric, ensuring empirical estimators remain consistent despite noise, discretization, and smoothing.
• The topological homeomorphism between measure and function spaces underpins robust estimation of key inequality indices such as Gini and Hoover coefficients.

The FRoM-W₁ framework formalizes a unified approach for representing probability measures via quantile functions and characterizing their topological and statistical properties under the Wasserstein-1 ($\mathrm{W}_1$) metric. It provides the theoretical foundation for the analysis of economic inequality indices, including the Gini and Hoover coefficients and the Lorenz curve, with rigorous convergence guarantees and homeomorphic relationships between measure space and function space. The framework underpins robust methodologies for both theoretical and empirical estimation of inequality, enabling the consistent reconstruction of these indices from finite data, noise, discretization, and kernel-based smoothing (Melot, 2024).

1. Quantile Function Representation of Measures

Let $\mu$ be a probability measure on $\mathbb{R}_+$ with finite, nonzero mean $m_\mu = \int_0^1 Q_\mu(p) dp$, where $Q_\mu$ is the quantile function $Q_\mu:[0,1)\to\mathbb{R}_+$. The quantile function uniquely determines the underlying distribution $\mu$ and encapsulates all information necessary for the computation of concentration indices such as the Lorenz curve, Gini, and Hoover coefficients.

The Wasserstein-1 distance between two measures $\mu,\nu$ with quantile functions $Q_\mu, Q_\nu$ is defined by

$W_1(\mu, \nu) = \int_0^1 |Q_\mu(p) - Q_\nu(p)| dp,$

establishing an explicit $L^1$ metric on the space of quantile functions. This quantile-based representation is central to the FRoM-W₁ framework, facilitating both algebraic manipulations and convergence proofs.

2. Lorenz Curve Geometry and Alternative Definitions of Inequality Indexes

The Lorenz curve for $\mu$ is constructed from its quantile function:

$L_\mu(p) = \frac{1}{m_\mu} \int_0^p Q_\mu(t) dt,$

which quantifies the cumulative proportion of total wealth owned by the lowest $p$-fraction of the population. $L_\mu$ is continuous, nondecreasing, convex, scale-invariant, and satisfies $L_\mu(0)=0$, $L_\mu(1)=1$, $L_\mu(p)\le p$.

Three principal forms are available for Gini ($G$) and Hoover ($H$) coefficients:

• Mean–absolute–difference form: $G(\mu) = \mathbb{E}[|X-X'|]/(2m_\mu)$, $H(\mu)=\mathbb{E}[|X-m_\mu|]/(2m_\mu)$ for i.i.d. $X,X'\sim\mu$.
• Quantile–$L^1$ form: $$G(\mu)=\frac{1}{2m_\mu}\int_0^1 \int_0^1|Q_\mu(p)-Q_\mu(q)|dp dq$$, $H(\mu)=\frac{1}{2m_\mu}\int_0^1|Q_\mu(p)-m_\mu| dp$.
• Lorenz–area form: $G(\mu)=1-2\int_0^1 L_\mu(p)dp$ (the area between $y=p$ and $L_\mu$), $H(\mu)=\max_{p\in[0,1]}[p-L_\mu(p)]$ (maximal vertical gap).

These equivalences, including pushforward and geometric interpretations, enable flexible statistical estimation and theoretical analysis.

3. Convergence Properties and Consistency Under $\mathrm{W}_1$ Metric

The FRoM-W₁ framework establishes the following equivalence for sequences $\mu_n,\mu_\infty$:

• $W_1(\mu_n,\mu_\infty)\to 0$
• $Q_{\mu_n}\to Q_{\mu_\infty}$ in $L^1([0,1])$
• $L_{\mu_n}\to L_{\mu_\infty}$ uniformly on $[0,1]$ and $m_{\mu_n}\to m_{\mu_\infty}$

Uniform convergence of Lorenz curves and Gini/Hoover coefficients underlies the consistency of empirical estimation from random samples, quantile discretizations, or kernel-smoothed reconstructions. This is achieved via Dini's lemma (leveraging convexity and monotonicity) and scrutiny of the left-derivative correspondence between quantile and Lorenz functions.

4. Topological Structure: Functional Homeomorphism

A central finding is the homeomorphism between the space of probability measures $(M, W_1)$ and the product space $$(C([0,1],\mathbb{R})_+, \| \cdot \|_\infty) \times (\mathbb{R}_+, |\cdot|)$$, realized by the mapping

$\Phi: \mu \mapsto (L_\mu, m_\mu),$

where $C([0,1],\mathbb{R})_+$ denotes continuous, convex, nondecreasing functions with $L(0)=0$, $L(1)=1$. This topological characterization permits the transfer of metric, continuity, and compactness results from measure theory to functional analysis, streamlining analytical tasks such as uniform convergence and functional approximation.

5. Statistical Implications: Robust Estimation and Perturbation Analysis

Empirical and theoretical consistency results follow directly from $\mathrm{W}_1$-based convergence. In particular:

• Sampling consistency: For empirical measures $\mu_n$ of i.i.d. samples from $\mu$, $W_1(\mu_n, \mu)\to 0$ almost surely, yielding $L_{\mu_n}\to L_\mu$, $G(\mu_n)\to G(\mu)$, $H(\mu_n)\to H(\mu)$.
• Perturbation regimes: Vanishing additive noise, quantile-grid approximations, and kernel-density estimates all preserve uniform convergence as information increases, subject to bounded integrability and vanishing bandwidth.
• Weaker convergence: Weak convergence plus first-moment convergence (Vitali–Scheffé) imply $L_{\mu_n}\to L_\mu$ pointwise; recovery of full uniform convergence requires uniform integrability or continuity at $p=1$.

This statistical robustness supports practical and theoretical implementations across sampling, smoothing, and finite-data estimation protocols.

6. Summary and Foundations of the FRoM-W₁ Framework

The FRoM-W₁ ("Functional Representation of Measures under $\mathrm{W}_1$") framework distills the core principles:

• Quantile-function representation encodes all concentration indices and allows direct quantification under the Wasserstein-1 metric.
• Lorenz-curve geometry abstracts inequality into continuous convex function space, supporting analysis and comparison via functional metrics.
• The $\mathrm{W}_1$ metric unifies weak and first-moment convergence, rendering statistical estimators for Gini, Hoover, and Lorenz indices robust with respect to sampling, smoothing, and perturbation.

This approach establishes a topologically and statistically coherent methodology for measuring and estimating economic inequality and related phenomena, with direct theoretical guarantees and empirical consistency results (Melot, 2024). A plausible implication is that analogous functional representations could be leveraged for other domains requiring the convergence of distributional indices under partial, noisy, or finite-data settings.