From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function

Cirillo, Pasquale; Fontanari, Andrea

doi:10.3390/math13132047

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From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function

by

Pasquale Cirillo

^1,*

and

Andrea Fontanari

²

¹

ZHAW School of Management and Law, Theaterstrasse 17, 8401 Winterthur, Switzerland

²

Optiver BV, Strawinskylaan 3095, 1077ZX Amsterdam, The Netherlands

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(13), 2047; https://doi.org/10.3390/math13132047

Submission received: 26 May 2025 / Revised: 18 June 2025 / Accepted: 19 June 2025 / Published: 20 June 2025

(This article belongs to the Special Issue Advanced Statistical Applications in Financial Econometrics)

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Abstract

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 M 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

λ_{U}

.

Keywords: Lorenz curve; pickands dependence function; extreme-value copula; inequality measures; tail dependence

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MDPI and ACS Style

Cirillo, P.; Fontanari, A. From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function. Mathematics 2025, 13, 2047. https://doi.org/10.3390/math13132047

AMA Style

Cirillo P, Fontanari A. From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function. Mathematics. 2025; 13(13):2047. https://doi.org/10.3390/math13132047

Chicago/Turabian Style

Cirillo, Pasquale, and Andrea Fontanari. 2025. "From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function" Mathematics 13, no. 13: 2047. https://doi.org/10.3390/math13132047

APA Style

Cirillo, P., & Fontanari, A. (2025). From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function. Mathematics, 13(13), 2047. https://doi.org/10.3390/math13132047

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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From Inequality to Extremes and Back: A Lorenz Representation of the Pickands Dependence Function

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