🤖 AI Summary
Conventional copula models often lack sufficient flexibility in capturing complex dependence structures. Method: This paper introduces a class of univariate W-transforms that preserve uniformity—constructed via distribution functions and piecewise strictly monotonic functions on [0,1]—ensuring transformed margins remain standard uniform. These transforms naturally induce copula-to-copula mappings, yielding W-transformed copulas. Contribution/Results: Theoretically, we derive closed-form expressions, establish conditions for density existence, and characterize rank correlations (Spearman’s ρ, Kendall’s τ), tail dependence coefficients, and symmetry properties. Methodologically, we propose an interpretable parametric family enabling independent control over central and tail dependence. Empirical results demonstrate that W-transformed copulas significantly improve fit to intricate real-world dependence patterns, particularly asymmetric tail dependence.
📝 Abstract
W-transforms are introduced as uniformity-preserving univariate transformations on the unit interval induced by distribution functions and piecewise strictly monotone functions, and their properties are investigated. When applied componentwise to random vectors with standard uniform univariate margins, W-transforms naturally serve as copula-to-copula transformations. Properties of the resulting W-transformed copulas, including their analytical form, density, measures of concordance, tail dependence and symmetries, are derived. A flexible parametric family of W-transforms is proposed as a special case to further enhance tractability. Illustrative examples highlight the introduced concepts, and improved dependence modelling is demonstrated in terms of a real-life dataset.