Inference for overparametrized hierarchical Archimedean copulas

TL;DR

This paper develops statistical tests and analytical tools for inference in overparametrized hierarchical Archimedean copulas, addressing model selection and overfitting issues in complex multivariate dependence modeling.

Contribution

It introduces asymptotic stochastic representations for maximum likelihood estimators and likelihood ratio tests for HAC structure hypotheses, filling a gap in inference methods.

Findings

Provided asymptotic stochastic representations for MLEs in HACs

Formulated likelihood ratio tests for structural hypotheses

Derived analytical derivatives for two-level HACs based on Clayton and Gumbel generators

Abstract

Hierarchical Archimedean copulas (HACs) are multivariate uniform distributions constructed by nesting Archimedean copulas into one another, and provide a flexible approach to modeling non-exchangeable data. However, this flexibility in the model structure may lead to over-fitting when the model estimation procedure is not performed properly. In this paper, we examine the problem of structure estimation and more generally on the selection of a parsimonious model from the hypothesis testing perspective. Formal tests for structural hypotheses concerning HACs have been lacking so far, most likely due to the restrictions on their associated parameter space which hinders the use of standard inference methodology. Building on previously developed asymptotic methods for these non-standard parameter spaces, we provide an asymptotic stochastic representation for the maximum likelihood estimators of (potentially) overparametrized HACs, which we then use to formulate a likelihood ratio test for certain common structural hypotheses. Additionally, we also derive analytical expressions for the first- and second-order partial derivatives of two-level HACs based on Clayton and Gumbel generators, as well as general numerical approximation schemes for the Fisher information matrix.