Semiparametric Gaussian copulas: normal scores estimator in high dimensions, and covariate-adjusted estimation

Yue Zhao now is Postdoctoral fellow at KU Leuven. He got Ph.D in statistics from Cornell University in 2015, and Ph.D in Physics from Princeton University in 2010. His research interests includes Copula method, survival analysis, high-dimensional statistical inference, empirical process.

The (semiparametric) Gaussian copula model consists of distributions that have dependence structure described by Gaussian copulas but that have arbitrary marginals.  A Gaussian copula is in turn determined by an Euclidean parameter called the copula correlation matrix $R$.  The Gaussian copula model has been intensively studied by both the high-dimensional statistics community and the traditional, fixed-dimensional asymptotics community.  In this talk we study two aspects of the normal scores (rank correlation coefficient) estimator of $R$.  In the first half of this talk, we consider the normal scores estimator in high dimensions.  It is well known that in fixed dimensions, the normal scores estimator is the optimal estimator of $R$, i.e., it has the smallest asymptotic covariance.  Curiously though, in high dimensions, nowadays the preferred estimators of $R$ are usually based on Kendall's tau or Spearman's rho.  We show that the normal scores estimator in fact remains the optimal estimator of $R$ in high dimensions.  In the second half of this talk, we investigate the inference of $R$ when the sample from the copula is perturbed by a covariate, and thus only indirectly observed as the response in a linear regression.  To remove the contamination by the covariate, we estimate the copula sample as the residuals from the linear regression based on a preliminary estimator of the coefficient matrix.  Then, we consider the normal scores estimator based on the residual ranks instead of the usual but unobservable oracle ranks.  We show that the residual-based normal scores estimator is asymptotically equivalent to its oracle counterpart, and provide explicit rate of convergence.