Estimating Gaussian Copulas with Missing Data

Estimating the joint distribution with data Missing At Random (MAR) is a hard task. Usually, one applies strictly parametric methods, mostly relying on members of the exponential family such as the multivariate normal distribution. Its parameters can be determined by the Expectation Maximization (EM) algorithm ([1]). However, the misspecification error in the case of non-Gaussian data might be considerable. We can extend the normality assumption and assume a Gaussian copula model, liberating us from restrictions on the shape of the marginals. Figure 1 is based on such a distribution for the bivariate case. Here, the green lines show the underlying marginal cumulative distribution functions for the first (left) and the second component (right) of a two-dimensional random vector (X1, X2), respectively. We then generated n 100 observations from (X1, X2) and calculated the corresponding empirical cumulative distribution functions (ecdf) (orange lines). To assess the influence of missings, we artificially chose some values to be Missing At Random (MAR) and recalculated the ecdfs based on the observed datapoints only (blue lines).