From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function
We establish a correspondence between Lorenz curves and Pickands dependence functions, thereby reframing the construction of any bivariate extreme-value copula as an inequality problem. We discuss the conditions under which a Lorenz curve generates a closed-form Pickands model, considerably expandin...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/13/2047 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849319831903404032 |
|---|---|
| author | Pasquale Cirillo Andrea Fontanari |
| author_facet | Pasquale Cirillo Andrea Fontanari |
| author_sort | Pasquale Cirillo |
| collection | DOAJ |
| description | We establish a correspondence between Lorenz curves and Pickands dependence functions, thereby reframing the construction of any bivariate extreme-value copula as an inequality problem. We discuss the conditions under which a Lorenz curve generates a closed-form Pickands model, considerably expanding the small set of tractable parametrizations currently available. Furthermore, the Pickands measure-generating function <i>M</i> can be written explicitly in terms of the quantile function underlying the Lorenz curve, providing a constructive route to model specification. Finally, classical inequality indices like the Gini coincide with scale-free, rotation-invariant indices of global upper-tail dependence, thereby complementing local coefficients such as the upper tail dependence index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>λ</mi><mi>U</mi></msub></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-2409d2d8ad5c4dc39e49dba40fcd7591 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-2409d2d8ad5c4dc39e49dba40fcd75912025-08-20T03:50:17ZengMDPI AGMathematics2227-73902025-06-011313204710.3390/math13132047From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence FunctionPasquale Cirillo0Andrea Fontanari1ZHAW School of Management and Law, Theaterstrasse 17, 8401 Winterthur, SwitzerlandOptiver BV, Strawinskylaan 3095, 1077ZX Amsterdam, The NetherlandsWe establish a correspondence between Lorenz curves and Pickands dependence functions, thereby reframing the construction of any bivariate extreme-value copula as an inequality problem. We discuss the conditions under which a Lorenz curve generates a closed-form Pickands model, considerably expanding the small set of tractable parametrizations currently available. Furthermore, the Pickands measure-generating function <i>M</i> can be written explicitly in terms of the quantile function underlying the Lorenz curve, providing a constructive route to model specification. Finally, classical inequality indices like the Gini coincide with scale-free, rotation-invariant indices of global upper-tail dependence, thereby complementing local coefficients such as the upper tail dependence index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>λ</mi><mi>U</mi></msub></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/13/13/2047Lorenz curvepickands dependence functionextreme-value copulainequality measurestail dependence |
| spellingShingle | Pasquale Cirillo Andrea Fontanari From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function Mathematics Lorenz curve pickands dependence function extreme-value copula inequality measures tail dependence |
| title | From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function |
| title_full | From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function |
| title_fullStr | From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function |
| title_full_unstemmed | From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function |
| title_short | From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function |
| title_sort | from inequality to extremes and back a lorenz representation of the pickands dependence function |
| topic | Lorenz curve pickands dependence function extreme-value copula inequality measures tail dependence |
| url | https://www.mdpi.com/2227-7390/13/13/2047 |
| work_keys_str_mv | AT pasqualecirillo frominequalitytoextremesandbackalorenzrepresentationofthepickandsdependencefunction AT andreafontanari frominequalitytoextremesandbackalorenzrepresentationofthepickandsdependencefunction |