Testing independence based on Bernstein empirical copula and copula density

ABSTRACT

In this paper we provide three nonparametric tests of independence between continuous random variables based on the Bernstein copula distribution function and the Bernstein copula density function. The first test is constructed based on a Cramér-von Mises divergence-type functional based on the empirical Bernstein copula process. The two other tests are based on the Bernstein copula density and use Cramér-von Mises and Kullback–Leibler divergence-type functionals, respectively. Furthermore, we study the asymptotic null distribution of each of these test statistics. Finally, we consider a Monte Carlo experiment to investigate the performance of our tests. In particular we examine their size and power which we compare with those of the classical nonparametric tests that are based on the empirical distribution function.

KEYWORDS:

• Bernstein empirical copula
• copula density
• Cramér–von Mises statistic
• Kullback–Leibler divergence-type
• independence test

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. For more details on copula theory, the readers are referred to an excellent book by Nelsen (2006).

Additional information

Funding

T. Bouezmarni and F. C. Lemyre gratefully acknowledge the research support of the Natural Sciences and Engineering Research Council of Canada [grant 402521-2013]. A. Taamouti gratefully acknowledge the research support from the Spanish MINECO [grants ECO2013-46395 and ECO2016-78652].