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, semiparametri- cally efficient estimator is proposed for the Euclidean copula param- eter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient 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 ef- ficiency of the pseudo-likelihood estimator. For correlation matrices structured according to a factor model, the pseudo-likelihood estima- tor turns out to be semiparametrically efficient. On the other hand, for Toeplitz correlation matrices, the asymptotic relative efficiency of the pseudo-likelihood estimator with respect to our one-step esti- mator can be as low as 20%. These findings are confirmed by Monte Carlo simulations