A proper scoring rule for minimum information copulas

TL;DR

This paper introduces the conditional Kullback-Leibler score, a proper scoring rule for minimum information copulas that circumvents the difficult computation of normalizing functions, enabling consistent and efficient estimation.

Contribution

The paper proposes a novel proper scoring rule for minimum information copulas that avoids normalizing function computation, facilitating easier and consistent estimation.

Findings

The score is strictly proper for copula densities.

The estimator is asymptotically consistent.

The score is convex and amenable to gradient optimization.

Abstract

Multi-dimensional distributions whose marginal distributions are uniform are called copulas. Among them, the one that satisfies given constraints on expectation and is closest to the independent distribution in the sense of Kullback-Leibler divergence is called the minimum information copula. The density function of the minimum information copula contains a set of functions called the normalizing functions, which are often difficult to compute. Although a number of proper scoring rules for probability distributions having normalizing constants such as exponential families are proposed, these scores are not applicable to the minimum information copulas due to the normalizing functions. In this paper, we propose the conditional Kullback-Leibler score, which avoids computation of the normalizing functions. The main idea of its construction is to use pairs of observations. We show that the proposed score is strictly proper in the space of copula density functions and therefore the estimator derived from it has asymptotic consistency. Furthermore, the score is convex with respect to the parameters and can be easily optimized by the gradient methods.