Rethinking Conventional Inequality Indices: Information Boundaries, Multifractal Spectrum Width, and Spatial Structural Information
- 1College of Finance and Mathematics, West Anhui University, Lu’an 237012, China
Abstract
Regional inequality is commonly measured with indices (e.g., the Gini and Theil indices) that depend only on the multiset of values and are therefore permutation-invariant: they cannot distinguish configurations differing solely in how the same values are ordered. This paper formalises that limitation as an information boundary and shows that the multifractal spectrum width (Δα) carries qualitatively distinct information. Four propositions are established: (1) with group membership fixed, the Gini, Theil, generalised entropy and Dagum Gini indices are permutation-invariant; (2) Δα is not, and can move in either direction under permutation; (3) Δα still varies appreciably when permutation-invariant indices are held constant, so the two classes are not informationally equivalent; and (4) controlled synthetic fields serve as a proof of concept linking Δα to multiscale heterogeneity. Permutation invariance secures the objectivity and comparability of inequality indices, but it also delineates their information boundary: Δα carries structural information that these indices cannot, in principle, capture, making the two complementary rather than substitutable. A finite-sample baseline and the Multifractal Magnitude Index (MMI) yield an operational diagnostic protocol; because Δα is defined only relative to the ordering rule and the MF-DFA parameterisation, all comparisons are made within a single parameterisation.
Related
- What Paid Triangles Identify about Reserve Uncertainty: Information Boundaries in Loss Reserving
- A Sharp Fisher-Information Inequality for Boolean Pairs: Proof of the Amari--Kobayashi Conjecture
- An affine reverse Faber-Krahn inequality via information decay
- A Self-Coupling Gaussian Extremal Inequality for Information-Constrained Optimal Transport with Applications to Rate-Distortion-Perception
- A nested factor model for equity markets: reconciling multifractal stock returns and rough index volatilities
BibTeX
@article{sun2026rethinking,
title = {{Rethinking Conventional Inequality Indices: Information Boundaries, Multifractal Spectrum Width, and Spatial Structural Information}},
author = {Luyun Sun and Benda Zhou},
journal = {Fractal and Fractional},
year = {2026},
month = oct,
doi = {10.3390/fractalfract10110764},
url = {https://www.mdpi.com/2504-3110/10/11/764},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Rethinking Conventional Inequality Indices: Information Boundaries, Multifractal Spectrum Width, and Spatial Structural Information
AU - Luyun Sun
AU - Benda Zhou
PY - 2026
DA - 2026/10/30
JO - Fractal and Fractional
DO - 10.3390/fractalfract10110764
UR - https://www.mdpi.com/2504-3110/10/11/764
AB - Regional inequality is commonly measured with indices (e.g., the Gini and Theil indices) that depend only on the multiset of values and are therefore permutation-invariant: they cannot distinguish configurations differing solely in how the same values are ordered. This paper formalises that limitation as an information boundary and shows that the multifractal spectrum width (Δα) carries qualitatively distinct information. Four propositions are established: (1) with group membership fixed, the Gini, Theil, generalised entropy and Dagum Gini indices are permutation-invariant; (2) Δα is not, and can move in either direction under permutation; (3) Δα still varies appreciably when permutation-invariant indices are held constant, so the two classes are not informationally equivalent; and (4) controlled synthetic fields serve as a proof of concept linking Δα to multiscale heterogeneity. Permutation invariance secures the objectivity and comparability of inequality indices, but it also delineates their information boundary: Δα carries structural information that these indices cannot, in principle, capture, making the two complementary rather than substitutable. A finite-sample baseline and the Multifractal Magnitude Index (MMI) yield an operational diagnostic protocol; because Δα is defined only relative to the ordering rule and the MF-DFA parameterisation, all comparisons are made within a single parameterisation.
SN - 2504-3110
ER -