An overview of the goodness-of-fit test problem for copulas

TL;DR

This paper reviews various goodness-of-fit tests for copulas, discussing their theoretical foundations, practical implementations, and performance in different scenarios, including time-dependent and specific copula families.

Contribution

It provides a comprehensive overview of existing omnibus goodness-of-fit procedures for copulas, highlighting their theoretical properties and practical effectiveness.

Findings

Summarizes key goodness-of-fit tests for copulas.

Discusses asymptotic distribution issues and p-value calculations.

Evaluates practical performance of various methods.

Abstract

We review the main "omnibus procedures" for goodness-of-fit testing for copulas: tests based on the empirical copula process, on probability integral transformations, on Kendall's dependence function, etc, and some corresponding reductions of dimension techniques. The problems of finding asymptotic distribution-free test statistics and the calculation of reliable p-values are discussed. Some particular cases, like convenient tests for time-dependent copulas, for Archimedean or extreme-value copulas, etc, are dealt with. Finally, the practical performances of the proposed approaches are briefly summarized.