Extreme points of Lorenz and ROC curves with applications to inequality analysis
EntityUAM. Departamento de Matemáticas
10.1016/j.jmaa.2022.126335Journal of Mathematical Analysis and Applications 514.2 (2022): 126335
ISSN0022-247X (print); 1096-0813 (online)
Funded byA. Baíllo and J. Cárcamo are supported by the Spanish MCyT grant PID2019-109387GB-I00. C. MoraCorral is supported by the Spanish MCyT grant MTM2017-85934-C3-2-P
ProjectGobierno de España. PID2019-109387GB-I00; Gobierno de España. MTM2017-85934-C3-2-P
SubjectsLorenz Curve; Decomposition; Inequality Indices; Matemáticas
Rights© 2022 The Authors
Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
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
Google Scholar:Baíllo Moreno, Amparo - Cárcamo Urtiaga, Javier - Mora Corral, Carlos
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