Smooth test for equality of copulas

TL;DR

This paper introduces a smooth, data-driven test for comparing multiple copulas simultaneously, effectively identifying differences even when densities are undefined, with demonstrated success on simulated and real data.

Contribution

The paper presents a novel smooth test for multiple copulas using copula coefficients, applicable even without density functions, and employs a two-step procedure for identifying significant differences.

Findings

Effective in detecting differences between copulas

Works without requiring copula densities

Validated through numerical and real data applications

Abstract

A smooth test to simultaneously compare $K$ copulas, where $K \geq 2$ is proposed. The $K$ observed populations can be paired, and the test statistic is constructed based on the differences between moment sequences, called copula coefficients. These coefficients characterize the copulas, even when the copula densities may not exist. The procedure employs a two-step data-driven procedure. In the initial step, the most significantly different coefficients are selected for all pairs of populations. The subsequent step utilizes these coefficients to identify populations that exhibit significant differences. To demonstrate the effectiveness of the method, we provide illustrations through numerical studies and application to two real datasets.