Nonparametric Regression and Error Covariance Function Estimation -- Beyond Short-Range Dependence

In nonparametric regression analysis, errors are possibly correlated in practice, and neglecting error correlation can undermine most bandwidth selection methods. When no prior knowledge or parametric form of the correlation structure is available in the random design setting, this issue has primarily been studied in the context of short-range dependent errors. When the data exhibits correlations that decay much more slowly, we introduce a special class of kernel functions and propose a procedure for selecting bandwidth in kernel-based nonparametric regression, using local linear regression as an example. Additionally, we provide a nonparametric estimate of the error covariance function, supported by theoretical results. Our simulations demonstrate significant improvements in estimating the nonparametric regression and error covariance functions, particularly in scenarios beyond short-range dependence. The practical application of our procedure is illustrated through the analysis of three datasets: cardiovascular disease mortality, life expectancy, and colon and rectum cancer mortality in the Southeastern United States.