Semiparametric Gaussian Copula Models : Geometry and Efficient Rank-Based Estimation

For multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically effi cient estimator is proposed for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the e fficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize adaptivity of the model with respect to the unknown marginal distributions and of efficiency of the pseudo-likelihood estimator. For correlation matrices structured according to a factor model, the pseudo-likelihood estimator turns out to be semiparametrically effi cient. On the other hand, for Toeplitz correlation matrices, the asymptotic relative effi ciency of the pseudo-likelihood estimator with respect to our one-step estimator can be as low as 20%. These findings are confirmed by Monte Carlo simulations