Properties of stepwise parameter estimation in high-dimensional vine copulas

TL;DR

This work develops a theory for stepwise parameter estimation in high-dimensional vine copulas where the number of parameters $p_n$ grows with the sample size $n$. It proves consistency and asymptotic normality of the stepwise MLE under diverging $p_n$ for known margins, parametric margins, and nonparametric margins, including truncated vines, with results applicable to general sequential estimation problems. The theoretical conditions are complemented by numerical validation and extensive simulations across Gaussian, Gumbel, and Student's t vines, illustrating practical feasibility and highlighting scenarios where estimation is challenging (notably D-vines and slow decay to independence). The findings inform high-dimensional vine modeling and provide guidance on margin treatment, truncation, and vine-structure choices, with code available for replication.

Abstract

The increasing use of vine copulas in high-dimensional settings, where the number of parameters is often of the same order as the sample size, calls for asymptotic theory beyond the traditional fixed-$p$, large-$n$ framework. We establish consistency and asymptotic normality of the stepwise maximum likelihood estimator for vine copulas when the number of parameters diverges as $n \to \infty$. Our theoretical results cover both parametric and nonparametric estimation of the marginal distributions, as well as truncated vines, and are also applicable to general estimation problems, particularly other sequential procedures. Numerical experiments suggest that the derived assumptions are satisfied if the pair copulas in higher trees converge to independence copulas sufficiently fast. A simulation study substantiates these findings and identifies settings in which estimation becomes challenging. In particular, the vine structure strongly affects estimation accuracy, with D-vines being more difficult to estimate than C-vines, and estimates in Gumbel vines exhibit substantially larger biases than those in Gaussian vines.

Figures (10)

• Figure 1: Graphical representations of two R-vine structures in four dimensions: D-vine (a) and C-vine (b). Each edge in the three trees denotes a pair copula.
• Figure 2: Estimated values for the validation of \ref{['A:curvature']} for Gaussian vines. The assumption is satisfied if the estimates are negative and bounded away from 0. The supremum over $\bm{\theta}$ is approximated by taking the maximum over $K=50$ values of $\bm{\theta}$. $\alpha_{j,n} = 1$ unless otherwise stated.
• Figure 3: Estimated values for $M_n^2$ and $D_n$ in \ref{['A:emp_proc']}. The supremum over $\bm{\theta}$ is approximated by taking the maximum over $K=30$ values of $\bm{\theta}$.
• Figure 4: Parameter estimation for Gaussian and Gumbel vines, mean maximum norm of estimation error for different proportions of $d$ and $n$. Parameterization: $\theta = \rho$ for Gaussian and $\text{Gumbel}(\theta + 1)$ for $\theta \ge 0$, $\text{Gumbel}_{90}(\theta-1)$ for $\theta <0$.
• Figure 5: Parameter estimation for Gaussian D-vine with $\theta_t^* = 1/\sqrt{t+1}$, mean maximum norm of estimation error with different normalizations.

Theorems & Definitions (25)

• Definition 1: R-vine specification Bedford2002
• Theorem 1: Consistency with known margins
• Theorem 2: Asymptotic normality with known margins
• Theorem 3: Consistency with parametric estimation of margins
• Theorem 4: Asymptotic normality with parametric estimation of margins
• Theorem 5: Consistency with nonparametric estimation of margins
• Theorem 6: Asymptotic normality with nonparametric estimation of margins
• Proposition 1
• proof
• Lemma 1