Bayesian estimation of a bivariate copula using the Jeffreys prior

A bivariate distribution with continuous margins can be uniquely decomposed via a copula and its marginal distributions. We consider the problem of estimating the copula function and adopt a Bayesian approach. On the space of copula functions, we construct a finite-dimensional approximation subspace that is parametrized by a doubly stochastic matrix. A major problem here is the selection of a prior distribution on the space of doubly stochastic matrices also known as the Birkhoff polytope. The main contributions of this paper are the derivation of a simple formula for the Jeffreys prior and showing that it is proper. It is known in the literature that for a complex problem like the one treated here, the above results are difficult to obtain. The Bayes estimator resulting from the Jeffreys prior is then evaluated numerically via Markov chain Monte Carlo methodology. A rather extensive simulation experiment is carried out. In many cases, the results favour the Bayes estimator over frequentist estimators such as the standard kernel estimator and Deheuvels’ estimator in terms of mean integrated squared error.

💡 Research Summary

The paper tackles the problem of estimating a bivariate copula function within a Bayesian framework. Recognizing that any continuous‑margin bivariate distribution can be uniquely expressed as a copula combined with its marginal distributions, the authors avoid direct non‑parametric estimation of the copula—an approach that suffers from high dimensionality and intricate constraints—by introducing a finite‑dimensional approximation. Specifically, they parametrize the copula through an n × n doubly stochastic matrix (all rows and columns sum to one), which lives on the Birkhoff polytope. Each vertex of this polytope corresponds to a permutation matrix, and the free parameters amount to (n‑1)², providing a flexible yet tractable representation of the dependence structure.

A central methodological hurdle is the choice of a prior distribution on the Birkhoff polytope. Conventional priors such as uniform or Dirichlet are either undefined or computationally intractable due to the polytope’s boundary complexity. The authors therefore derive the Jeffreys prior, which is proportional to the square root of the determinant of the Fisher information matrix. By incorporating the row‑ and column‑sum constraints via Lagrange multipliers, they obtain a closed‑form expression for the prior density: \