The Empirical Beta Copula

TL;DR

This paper introduces the empirical beta copula, a new nonparametric estimator for copulas that improves bias and variance properties over existing methods, supported by theoretical proofs and simulation results.

Contribution

It defines the empirical beta copula as a genuine copula derived from rearranged uniform variates, and establishes its asymptotic equivalence to the empirical copula under weaker assumptions.

Findings

Empirical beta copula outperforms empirical copula in bias and variance.

Theoretical proof that the empirical beta copula is a genuine copula.

Simulation shows improved finite-sample performance, especially in bias.

Abstract

Given a sample from a 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. Stat. Plan. Infer. 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.