We consider the fixed-design regression model with long-range dependent normal errors and show that the finite-dimensional distributions of the properly normalized Gasser-Muller and Priestley-Chao estimators of the regression function converge to those of a white noise process. Furthermore, the distributions of the suitably renormalized maximal deviations over an increasingly finer grid converge to the Gumbel distribution. These results contrast with our previous findings for the corresponding problem of estimating the marginal density of long-range dependent stationary sequences.
"Nonparametric Regression Under Long-Range Dependent Normal Errors." Ann. Statist. 23 (3) 1000 - 1014, June, 1995. https://doi.org/10.1214/aos/1176324633