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Gaussian processes that can be decomposed into a smooth mean function and astationary autocorrelated noise process are considered and a fully automaticnonparametric method to simultaneous estimation of mean and auto-covariancefunctions of such processes is developed. Our empirical Bayes approach isdata-driven, numerically efficient and allows for the construction ofconfidence sets for the mean function. Performance is demonstrated insimulations and real data analysis. The method is implemented in the R packageeBsc that accompanies the paper.

Adaptive Non-parametric Estimation of Mean and Autocovariance in Regression with Dependent Errors