[Paper Review] Likelihood inference for Archimedean copulas

Marius Hofert, Martin Mächler+1 more|arXiv (Cornell University)|Aug 30, 2011

Financial Risk and Volatility Modeling31 references3 citations

TL;DR

This paper derives explicit functional forms for generator derivatives of well-known Archimedean copulas, enabling precise and efficient maximum-likelihood estimation even in high dimensions (up to d=100). The key contribution is a numerically stable, open-source R implementation in the nacopula package, which achieves decreasing root mean squared error with increasing dimension and enables likelihood-based confidence intervals and multi-parameter estimation.

ABSTRACT

Explicit functional forms for the generator derivatives of well-known one-parameter Archimedean copulas are derived. These derivatives are essential for likelihood inference as they appear in the copula density, conditional distribution functions, or the Kendall distribution function. They are also required for several asymmetric extensions of Archimedean copulas such as Khoudraji-transformed Archimedean copulas. Access to the generator derivatives makes maximum-likelihood estimation for Archimedean copulas feasible in terms of both precision and run time, even in large dimensions. It is shown by simulation that the root mean squared error is decreasing in the dimension. This decrease is of the same order as the decrease in sample size. Furthermore, confidence intervals for the parameter vector are derived. Moreover, extensions to multi-parameter Archimedean families are given. All presented methods are implemented in the open-source R package nacopula and can thus easily be accessed and studied.

Motivation & Objective

• To overcome the computational challenges in deriving and evaluating generator derivatives for Archimedean copulas, especially in high dimensions.
• To enable accurate and efficient maximum-likelihood estimation (MLE) for Archimedean copulas by providing closed-form expressions for required derivatives.
• To extend MLE to multi-parameter Archimedean families, including strategies for initial interval selection and confidence interval construction.
• To demonstrate that root mean squared error (RMSE) decreases with increasing dimension, scaling similarly to sample size reduction.
• To provide a transparent, open-source implementation in the R package nacopula for reproducibility and practical use.

Proposed method

• Derives explicit, non-recursive functional forms for generator derivatives of standard Archimedean families (e.g., Gumbel, Clayton, GIG), crucial for copula density and distribution functions.
• Uses analytical expressions for derivatives to compute copula densities, conditional distributions, and Kendall distribution functions without relying on symbolic computation systems.
• Applies numerical optimization with carefully constructed initial intervals for likelihood maximization, especially for multi-parameter families like GIG copulas.
• Employs a two-stage algorithm to determine initial parameter rectangles based on estimated Kendall’s tau, ensuring convergence in high-dimensional settings.
• Constructs likelihood-based confidence intervals for parameter vectors and compares them with Fisher information-based intervals for the Clayton family.
• Implements all algorithms in the open-source R package nacopula, ensuring reproducibility and accessibility for researchers.

Experimental results

Research questions

• RQ1 Can explicit, numerically stable expressions for generator derivatives be derived for well-known Archimedean copulas in arbitrary dimensions?
• RQ2 Does maximum-likelihood estimation for Archimedean copulas remain precise and computationally feasible in high dimensions (d ≥ 10)?
• RQ3 How does the root mean squared error (RMSE) of parameter estimates scale with increasing dimension and sample size?
• RQ4 Can reliable initial intervals be constructed for likelihood optimization in multi-parameter Archimedean families?
• RQ5 Are likelihood-based confidence intervals for copula parameters comparable to information-based intervals in terms of coverage and accuracy?

Key findings

• The root mean squared error (RMSE) of parameter estimates decreases with increasing dimension d, and this decrease is of the same order as the decrease in sample size n, i.e., RMSE ∝ 1/√(nd).
• Maximum-likelihood estimation for Archimedean copulas is both precise and computationally efficient, even in high dimensions up to d=100, as demonstrated in prior work and confirmed here.
• For the GIG copula family, larger sample sizes are required to achieve low bias under weak concordance (low Kendall’s tau), especially when ν is small.
• Initial intervals for multi-parameter families (e.g., GIG) can be reliably constructed using an algorithm based on estimated Kendall’s tau, ensuring convergence in optimization.
• Likelihood-based confidence intervals for the parameter vector are feasible and comparable to Fisher information-based intervals, particularly for the Clayton family where Fisher information is analytically tractable.
• All methods are implemented and publicly available in the open-source R package nacopula, enabling full reproducibility and practical application by researchers.