Estimating checkerboard approximations with sample d-copulas

A novel characterization and new simple tests of multivariate independence using copulas

“…i d ) ∈ I d m . We now provide an efficient method to estimate the checkerboard approximation of order m, that we call the sample d-copula of order m. This methodology first appeared in [9] and was significantly improved in [11]. First, let X 1 , .…”

“…It can be shown, see [11], that C (m) n is always a d-copula and that it estimates C (m) . In fact, C (m) n is related to a type of kernel estimator of the true copula C when the density of C exists.…”

“…It is worth pointing out that the sample d-copula of order m is far easier to compute than the empirical copula, the Bernstein copulas and the beta empirical copulas, see [14] and [20], all of which have been used to estimate the true copula C. However, as shown in [11], in all of these cases we can obtain better approximations to the true copula C using the sample copula.…”

“…The use of a sample d-copula of order m (denoted C (m) n ) as an estimator of C (m) was studied in [11]. Once the order m has been chosen, the sample d-copula of order m can be thought of as a kind of kernel-based estimator of C. It turns out to be a good estimator of the copula C, even for moderate values of m, and improves on the empirical copula, the Bernstein copula and the beta empirical copula [11]. On the other hand, the sample copula can be easily computed even for moderate sample sizes.…”

“…First, we provide a novel characterization of independence of random vectors based on the checkerboard approximation to a multivariate copula.Using this result, we then propose a new family of tests of multivariate independence for continuous random vectors. The tests rely on estimating the checkerboard approximation by means of the sample copula recently introduced in [9] and improved in [11]. Such estimators have nice properties, including a Glivenko-Cantelli-type theorem that guarantees almost-sure uniform convergence to the checkerboard approximation.…”