The empirical beta copula

JOURNAL OF MULTIVARIATE ANALYSIS

卷:155

Article

Segers, Johan¹  Sibuya, Masaaki²  Tsukahara, Hideatsu³ 

[1] Catholic Univ Louvain, Inst Stat Biostat & Sci Actuarielles, Voie Roman Pays 20, B-1348 Louvain La Neuve, Belgium

[2] Keio Univ, Tokyo 108, Japan

[3] Seijo Univ, Fac Econ, Setagaya Ku, 6-1-20 Seijo, Tokyo 1578511, Japan

关键词: Copula;    Empirical copula;    Bernstein polynomial;    Empirical Bernstein copula;    Checkerboard copula;   

DOI  :  10.1016/j.jmva.2016.11.010

来源: Elsevier

【 摘 要 】

Given a sample from a continuous multivariate distribution F, the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of F. This idea can be regarded as a variant on Baker's [J. Multivariate Anal. 99 (2008) 2312-2327] copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size. Necessary and sufficient conditions are given for a Bernstein polynomial to be a copula. These imply that the empirical beta copula is a genuine copula. Furthermore, the empirical process based on the empirical Bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in Janssen, Swanepoel and Veraverbeke [J. Statist. Plann. Inference 142 (2012) 1189-1197]. A Monte Carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance. Compared with the empirical Bernstein copula with the smoothing rate suggested by Janssen et al., its finite-sample performance is still significantly better in several cases, especially in terms of bias. (C) 2016 Elsevier Inc. All rights reserved.

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