  We develop adaptive estimation and inference methods for high-dimensional
Gaussian copula regression that achieve the same performance without the
knowledge of the marginal transformations as that for high-dimensional linear
regression. Using a Kendall's tau based covariance matrix estimator, an
$\ell_1$ regularized estimator is proposed and a corresponding de-biased
estimator is developed for the construction of the confidence intervals and
hypothesis tests. Theoretical properties of the procedures are studied and the
proposed estimation and inference methods are shown to be adaptive to the
unknown monotone marginal transformations. Prediction of the response for a
given value of the covariates is also considered. The procedures are easy to
implement and perform well numerically. The methods are also applied to analyze
the Communities and Crime Unnormalized Data from the UCI Machine Learning
Repository.
