Extreme points of Lorenz and ROC curves with applications to inequality analysis
We find the extreme points of the set of convex functions ℓ : [0,1] → [0,1] with a fixed area and ℓ(0) = 0, ℓ(1) = 1. This collection is formed by Lorenz curves with a given value of their Gini index. The analogous set of concave functions can be viewed as Receiver Operating Characteristic (ROC) cur...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/707315 |
| Acceso en línea: | http://hdl.handle.net/10486/707315 https://dx.doi.org/10.1016/j.jmaa.2022.126335 |
| Access Level: | acceso abierto |
| Palabra clave: | Lorenz Curve Decomposition Inequality Indices Matemáticas |
| Sumario: | We find the extreme points of the set of convex functions ℓ : [0,1] → [0,1] with a fixed area and ℓ(0) = 0, ℓ(1) = 1. This collection is formed by Lorenz curves with a given value of their Gini index. The analogous set of concave functions can be viewed as Receiver Operating Characteristic (ROC) curves. These functions are extensively used in economics (inequality and risk analysis) and machine learning (evaluation of the performance of binary classifiers). We also compute the maximal L1-distance between two Lorenz (or ROC) curves with specified Gini coefficients. This result allows us to introduce a bidimensional index to compare two of such curves, in a more informative and insightful manner than with the usual unidimensional measures considered in the literature (Gini index or area under the ROC curve). The analysis of real income microdata illustrates the practical use of this proposed index in statistical inference |
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