The empirical copula process in high dimensions: Stute's representation and applications

TL;DR

This paper extends the empirical copula process to high-dimensional settings, demonstrating that low-dimensional margins can be approximated by a functional of the empirical process, with applications in independence testing and error control.

Contribution

It establishes the validity of Stute's representation in high dimensions and applies it to improve independence tests with bootstrap methods.

Findings

Stute's representation holds in high dimensions under weak smoothness assumptions.

Type-I error control achieved for pairwise independence tests beyond mutual independence.

Bootstrap critical values provide strong familywise error rate control.

Abstract

The empirical copula process, a fundamental tool for copula inference, is studied in the high dimensional regime where the dimension is allowed to grow to infinity exponentially in the sample size. Under natural, weak smoothness assumptions on the underlying copula, it is shown that Stute's representation is valid in the following sense: all low-dimensional margins of fixed dimension of the empirical copula process can be approximated by a functional of the low-dimensional margins of the standard empirical process, with the almost sure error term being uniform in the margins. The result has numerous potential applications, and is exemplary applied to the problem of testing pairwise stochastic independence in high dimensions, leading to various extensions of recent results in the literature: for certain test statistics based on pairwise association measures, type-I error control is obtained for models beyond mutual independence. Moreover, bootstrap-based critical values are shown to yield strong control of the familywise error rate for a large class of data generating processes.