Pattern-based tests for two-dimensional copulas

TL;DR

This paper develops pattern-based statistical tests for two-dimensional copulas, using a functional central limit theorem for pattern frequencies, applicable in nonparametric and parametric contexts.

Contribution

It introduces a new theoretical framework for pattern-based goodness-of-fit and two-sample tests for two-dimensional copulas, including bootstrap methods and implementation insights.

Findings

Established a functional central limit theorem for pattern frequencies.

Developed nonparametric goodness-of-fit and two-sample tests based on pattern procedures.

Provided simulation results validating the theoretical methods.

Abstract

In statistics permutations typically arise in the context of rank plots for two-dimensional data. Such plots can also be interpreted as discrete copulas. In discrete mathematics, typically in the context of the description of large (non-random) objects, two-dimensional copulas appear as limits of permutations and are then known as permutons if the topology refers to the convergence of pattern frequencies. We obtain a functional central limit theorem for such pattern frequencies in the context of two-dimensional random samples. The result serves as the basis for nonparametric goodness-of-fit tests, for two-sample tests, and for tests of symmetry. This includes a suitable variant of the bootstrap for obtaining critical values. Pattern-based procedures are also of interest in a parametric context. We consider two examples, the Farlie-Gumbel-Morgenstern class and a family of delay copulas. We discuss implementation aspects of the resulting procedures and we provide a simulation study that supplements the theoretical results in the nonparametric case.