Often of primary interest in the analysis of multivariate data are the copula parameters describing the dependence among the variables, rather than the univariate marginal distributions. Since the ranks of a multivariate dataset are invariant to changes in the univariate marginal distributions, rank-based procedures are natural candidates as semiparametric estimators of copula parameters. Asymptotic information bounds for such estimators can be obtained from an asymptotic analysis of the rank likelihood, i.e. the probability of the multivariate ranks. In this article, we obtain limiting normal distributions of the rank likelihood for Gaussian copula models. Our results cover models with structured correlation matrices, such as exchangeable, autoregressive and circular correlation, as well as unstructured correlation matrices. For all Gaussian copula models, the limiting distribution of the rank likelihood ratio is shown to be equal to that of a parametric likelihood ratio for an appropriately chosen multivariate normal model. This implies that the semiparameteric information bounds for rank-based estimators are the same as the information bounds for estimators based on the full data, and that the multivariate normal distributions are least favorable.
Keywords: copula model, local asymptotic normality, multivariate rank statistics, marginal likelihood, rank likelihood, transformation model.