Measuring multidimensional inequality: a new proposal based on the Fourier transform
Paolo Giudici, Emanuela Raffinetti, Giuseppe Toscani
TL;DR
This paper introduces a Fourier-transform-based framework for measuring multidimensional inequality, yielding a scaling-invariant index $\tau_n(F)$ that generalizes univariate Fourier-based Gini/Pietra measures. By incorporating a covariance-aware transform $\pmb\xi^*$, the authors connect the index to the Mahalanobis distance, enabling unit-free, component-wise comparisons in multi-attribute settings. They derive closed-form results for a two-point distribution and for multivariate Gaussians, the latter tying the index to the multivariate coefficient of variation $C_{VN}$ with a constant factor $1/(2\sqrt{e})$. The work also establishes convexity and convolution-related properties, and proposes multivariate extensions of Pietra and Gini, highlighting practical implications for economics and risk aggregation where multi-attribute inequality matters.
Abstract
Inequality measures are quantitative measures that take values in the unit interval, with a zero value characterizing perfect equality. Although originally proposed to measure economic inequalities, they can be applied to several other situations, in which one is interested in the mutual variability between a set of observations, rather than in their deviations from the mean. While unidimensional measures of inequality, such as the Gini index, are widely known and employed, multidimensional measures, such as Lorenz Zonoids, are difficult to interpret and computationally expensive and, for these reasons, are not much well known. To overcome the problem, in this paper we propose a new scaling invariant multidimensional inequality index, based on the Fourier transform, which exhibits a number of interesting properties, and whose application to the multidimensional case is rather straightforward to calculate and interpret.