  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.
