Statistics and Its Interface

Volume 15 (2022)

Number 4

Multivariate Bernstein Fréchet copulas

Pages: 527 – 547

DOI: https://dx.doi.org/10.4310/22-SII722

Authors

Zongkai Xie (Department of Financial Mathematics, Peking University, Beijing, China)

Fang Wang (School of Mathematical Sciences, Capital Normal University, Beijing, China)

Jingping Yang (LMEQF, Department of Financial Mathematics, Peking University, Beijing, China)

Nan Guo (China Bond Rating Company, Beijing, China)

Abstract

Finding joint copulas based on given bivariate margins is an interesting problem. It involves in obtaining the copula from the information of its bivariate marginal distributions. In this paper, we present a multivariate copula family called multivariate Bernstein Fréchet (BF) copulas. Each copula in the family is uniquely determined by its bivariate margins, the bivariate BF copulas. For this purpose, we first discuss properties of the bivariate BF copulas, including supermigrativity and $\mathrm{TP}_2$ properties. The advantages of bivariate BF copula are identified by comparing it with the bivariate Gaussian copula and the bivariate Fréchet copula. We show that a multivariate BF copula is uniquely determined by its marginal bivariate BF copulas, and methods to construct the multivariate BF copula are discussed. Numerical studies are carried out for displaying the advantages of multivariate BF copulas.

Keywords

copula construction, multivariate Bernstein Fréchet copulas, bivariate marginal copulas, parametric estimation

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The research of Xie and Yang were supported by the National Natural Science Foundation of China (Grant No. 12071016).

The research of Wang was supported by the National Natural Science Foundation of China (Grant No. 12171328) and Academy for Multidisciplinary Studies, Capital Normal University.

Received 13 December 2020

Accepted 3 January 2022

Published 4 March 2022