Estimators for Archimedean copulas in high dimensions

The performance of known and new parametric estimators for Archimedean copulas is investigated, with special focus on large dimensions and numerical difficulties. In particular, method-of-moments-like estimators based on pairwise Kendall’s tau, a multivariate extension of Blomqvist’s beta, minimum distance estimators, the maximum-likelihood estimator, a simulated maximum-likelihood estimator, and a maximum-likelihood estimator based on the copula diagonal are studied. Their performance is compared in a large-scale simulation study both under known and unknown margins (pseudo-observations), in small and high dimensions, under small and large dependencies, various different Archimedean families and sample sizes. High dimensions up to one hundred are considered for the first time and computational problems arising from such large dimensions are addressed in detail. All methods are implemented in the open source \R{} package \pkg{copula} and can thus be easily accessed and studied.

💡 Research Summary

The paper conducts a comprehensive comparative study of several parametric estimators for Archimedean copulas when the dimension of the data is large, up to one hundred variables. Six estimation strategies are examined: (i) a pairwise Kendall‑tau method‑of‑moments, (ii) a multivariate extension of Blomqvist’s beta, (iii) minimum‑distance estimators (MDE) based on Cramér‑von Mises or Kolmogorov‑Smirnov distances, (iv) the classical maximum‑likelihood estimator (MLE), (v) a simulated‑likelihood MLE that approximates the copula density by Monte‑Carlo sampling, and (vi) a diagonal‑based MLE that uses only the one‑dimensional distribution of the copula’s diagonal.

The authors first derive the explicit forms of the likelihood and its derivatives for the four most common Archimedean families (Clayton, Gumbel, Frank, Joe). They discuss the numerical difficulties that arise when the generator’s high‑order derivatives become extremely small or large, and propose log‑scale transformations, automatic differentiation, and careful initialization (using the method‑of‑moments estimates) to improve stability. For the simulated‑likelihood approach, importance sampling and adaptive sample‑size selection are introduced to keep the computational burden proportional to the dimension rather than to the full combinatorial complexity. The diagonal‑based estimator exploits the fact that the copula evaluated on the line (u,…,u) reduces to a univariate distribution, which can be evaluated analytically for all four families, yielding a fast and numerically robust alternative.

A large‑scale Monte‑Carlo simulation study is then performed. Sample sizes of n = 200, 500, 1 000 are considered, and dependence levels are varied to produce Kendall’s τ ≈ 0.2 (weak), 0.5 (moderate), and 0.8 (strong). Both the “known‑margin” scenario and the “pseudo‑observation” scenario (where margins are estimated non‑parametrically) are examined. For each combination of dimension (d = 5, 10, 20, 50, 100), copula family, dependence level, and sample size, the authors compute bias, mean‑squared error (MSE), and average CPU time for each estimator.

Key findings include:

• In low to moderate dimensions (d ≤ 20) the pairwise Kendall‑tau and multivariate Blomqvist estimators perform competitively in terms of MSE, but their computational cost grows quadratically with d, making them impractical for d ≥ 50.
• Minimum‑distance estimators are flexible and can capture complex dependence patterns, yet their reliance on numerical integration leads to higher MSE when the integration error is non‑negligible, especially for strong dependence.
• The classical MLE provides the lowest MSE when the likelihood can be evaluated accurately; however, for d ≥ 50 the required high‑order derivatives cause numerical under‑/overflow, even with log‑transformations.
• The simulated‑likelihood MLE achieves MSE comparable to the exact MLE but incurs substantially higher CPU times because of the Monte‑Carlo approximation; its advantage is only evident when the exact likelihood is unavailable or unstable.
• The diagonal‑based MLE emerges as the most computationally efficient method across all dimensions. Its MSE is slightly higher than the exact MLE for weak dependence but becomes comparable or better for moderate to strong dependence, and it remains numerically stable even at d = 100.

When margins are unknown and pseudo‑observations are used, all estimators experience a modest increase in bias and variance, yet the diagonal‑based MLE and the MDE retain relatively robust performance.

All algorithms have been implemented in the open‑source R package copula, with careful attention to memory management, parallel processing, and reproducible code. The authors provide detailed documentation and examples, enabling practitioners to apply these methods to high‑dimensional data sets without having to develop custom code.

In conclusion, the study demonstrates that for high‑dimensional Archimedean copulas the diagonal‑based maximum‑likelihood estimator offers the best trade‑off between statistical efficiency, computational speed, and numerical stability. Nevertheless, the choice of estimator should be guided by the specific scientific goal: if a full description of the multivariate dependence structure is required, minimum‑distance or simulated‑likelihood methods may be preferable despite their higher computational cost. The paper thus supplies a practical roadmap for researchers confronting high‑dimensional dependence modeling, and it enriches the statistical toolbox with robust, ready‑to‑use implementations.