An iterated $I$-projection procedure for solving the generalized minimum information checkerboard copula problem

TL;DR

This work proposes a generalization of the minimum information copula principle allowing the inclusion of additional constraints fixing certain higher-order margins of the copula and shows that the associated optimization problem has a unique solution under a natural condition.

Abstract

The minimum information copula principle initially suggested in \cite{MeeBed97} is a maximum entropy-like approach for finding the least informative copula, if it exists, that satisfies a certain number of expectation constraints specified either from domain knowledge or the available data. We first propose a generalization of this principle allowing the inclusion of additional constraints fixing certain higher-order margins of the copula. We next show that the associated optimization problem has a unique solution under a natural condition. As the latter problem is intractable in general we consider its version with all the probability measures involved in its formulation replaced by checkerboard approximations. This amounts to attempting to solve a so-called discrete $I$-projection linear problem. We then exploit the seminal results of \cite{Csi75} to derive an iterated procedure for solving the latter and provide theoretical guarantees for its convergence. The usefulness of the procedure is finally illustrated via numerical experiments in dimensions up to four with substantially finer discretizations than those encountered in the literature.

Figures (9)

• Figure 1: Left: one realization of a random sample of size $1000$ from a discrete random vector $(V_1,V_2)$ with support $\mathcal{X}$ in \ref{['eq:Xc']} and probability array $q^{[3 \times 144]}$ obtained with Procedure II when $n = 30$. Middle: one realization of a random sample of size $1000$ from $\check Q^{[3 \times 144]}$. Right: maximum absolute error as defined in Algorithm \ref{['algo:iterated:Iproj']} against the iteration number.
• Figure 2: Left: one realization of a random sample of size $1000$ from a discrete random vector $(V_1,V_2)$ with support $\mathcal{X}$ in \ref{['eq:Xc']} and probability array $q^{[3 \times 115]}$ obtained with Procedure II when $n = 300$. Middle: one realization of a random sample of size $1000$ from $\check Q^{[3 \times 115]}$. Right: maximum absolute error as defined in Algorithm \ref{['algo:iterated:Iproj']} against the iteration number.
• Figure 3: Maximum absolute error against the iteration number for Procedure I (left) and Procedure II (right) for the inconsistent Spearman's rho constraints in the trivariate case.
• Figure 4: Scatterplot matrix of a realization of a random sample of size $1000$ from the trivariate checkerboard copula returned by Procedure II for the experiment with $n=100$, $\alpha_{\{1,2\}} = 0.4$, $\alpha_{\{1,3\}} = 0.6$ and $\alpha_{\{2,3\}} = 0.8$.
• Figure 5: Scatterplot matrix of a realization of a random sample of size $1000$ from the trivariate checkerboard copula returned by Procedure II for the experiment with $n=100$ and $\alpha_{\{2,3\}} = 0.8$.

Theorems & Definitions (30)

• Definition 2.1: $I$-projection
• Definition 2.2: $I$-projection for probability arrays
• Remark 2.3
• Proposition 3.1
• Remark 3.2
• Remark 3.3
• Proposition 3.6
• Remark 3.7
• Remark 3.8
• Corollary 4.1