Determining the number of components within a copula family

Determine the number of mixture components supported within each copula family for the heterogeneous Clayton--Gumbel--Frank--Gaussian mixture framework.

Background

The theoretical analysis permits a fixed maximum number of mixture components and establishes identifiability for finite mixtures drawn from the specified copula dictionary. It does not resolve the model-selection problem of determining how many components are genuinely needed within a given family.

This issue is distinct from identifying which copula families contribute to the mixture. In overfitted same-family mixtures, the paper notes that redundant components may duplicate a true component and share its weight, so component-level behavior is not covered by the stated consistency results. A solution would address within-family order selection and the treatment of redundant components.

References

Two questions remain open for this dictionary: rates of contraction, which require stronger identifiability conditions \citep{Heinrich2018} and would quantify how fast absent families lose their weight, and the number of components within a family \citep{Rousseau2011}.

— Bayesian heterogeneous copula mixtures with nonparametric margins: consistency, identifiability and tail asymmetry in physical fitness data  (2610.08317 - Liu, 6 Oct 2026) in Section Discussion, final paragraph