A Tractable Family of Smooth Copulas with Rotational Dependence: Properties, Inference, and Application

TL;DR

This work introduces a tractable, interpretable family of copulas on $[0,1]^d$ generated by a univariate density on $[0,1]$, linking dependence strength and form to the smoothness of the generator and a signature that encodes rotational/reflectional structure. The core construction yields $c_f^{\bm{s}}(\bm{u}) = f\left(\bigoplus_{j=1}^d \widetilde{u}_j\right)$ with $\widetilde{u}_j = u_j^{1-s_j}(1-u_j)^{s_j}$ and wrapped-sum $\bigoplus$, and a simple sampling algorithm facilitates practical use. The authors develop both parametric and nonparametric inference (signature learning, generator estimation via MLE/KDE, and empirical copula theory) and demonstrate robustness through simulations and a neural-connectivity application where the model captures rotational dependence with competitive fit. They further show that multivariate estimation reduces to univariate problems, enabling scalable, rigorous analysis, and discuss extensions via hierarchical generators and circle-group isometries. Overall, this framework provides a flexible, well-founded approach to modeling angular dependence with transparent interpretability and solid statistical guarantees.

Abstract

We introduce a new family of copula densities constructed from univariate distributions on $[0,1]$. Although our construction is structurally simple, the resulting family is versatile: it includes both smooth and irregular examples, and reveals clear links between properties of the underlying univariate distribution and the strength, direction, and form of multivariate dependence. The framework brings with it a range of explicit mathematical properties, including interpretable characterizations of dependence and transparent descriptions of how rotational forms arise. We propose model selection and inference methods in parametric and nonparametric settings, supported by asymptotic theory that reduces multivariate estimation to well-studied univariate problems. Simulation studies confirm the reliable recovery of structural features, and an application involving neural connectivity data illustrates how the family can yield a better fit than existing models.

Figures (22)

• Figure 1: Bivariate copula densities corresponding to the piecewise constant generators $f_1$ and $f_{10}$, and the limiting triangular distribution $f_\infty.$
• Figure 2: Top row: bivariate copula densities corresponding to the beta generators $f_{\frac{3}{2},\frac{3}{2}}$, $f_{\frac{1}{2},\frac{1}{2}}$, and $f_{\frac{1}{2},\frac{3}{2}}$. Bottom row: different views of the same graphs.
• Figure 3: Bivariate copula densities corresponding to the mixture of truncated normals generator with signatures $\bm{0}$, $(0,1)$, $(1,0)$, and $\bm{1}$.
• Figure 4: From left to right, the normalized partial sums $h_1^*$, $h_{10}^*$ and $h_{100}^*$.
• Figure 5: Proportion of replicates with incorrect signature recovery by method (Kolmogorov--Smirnov versus Cramér--von Mises), sample size $n \in \{50, 100, 200, 500, 1000\}$, and dimension $d \in \{2,3,4,5\}$, across several generator families used in \ref{['sub:simpleexamples']}. The recovery rate reaches $100\%$ quickly and is largely insensitive to $d$ and the generator family.

Theorems & Definitions (43)

• Proposition 1
• Lemma 1
• proof
• Remark
• proof : Proof of \ref{['prop:Ccharacteriation']}
• Proposition 2
• proof
• Corollary 1
• proof
• Proposition 3