Semiparametric Gaussian copula models: Geometry and efficient rank-based estimation

Abstract

We propose, for multivariate Gaussian copula models with unknown margins and\nstructured correlation matrices, a rank-based, semiparametrically efficient\nestimator for the Euclidean copula parameter. This estimator is defined as a\none-step update of a rank-based pilot estimator in the direction of the\nefficient influence function, which is calculated explicitly. Moreover,\nfinite-dimensional algebraic conditions are given that completely characterize\nefficiency of the pseudo-likelihood estimator and adaptivity of the model with\nrespect to the unknown marginal distributions. For correlation matrices\nstructured according to a factor model, the pseudo-likelihood estimator turns\nout to be semiparametrically efficient. On the other hand, for Toeplitz\ncorrelation matrices, the asymptotic relative efficiency of the\npseudo-likelihood estimator can be as low as 20%. These findings are confirmed\nby Monte Carlo simulations. We indicate how our results can be extended to\njoint regression models.\n