Some results on random design regression with long memory errors and predictors

Abstract: This paper studies nonparametric regression with long memory (LRD) errors and predictors. First, we formulate general conditions which guarantee the standard rate of convergence for a nonparametric kernel estimator. Second, we calculate the Mean Integrated Squared Error (MISE). In particular, we show that LRD of errors may influence MISE. On the other hand, an estimator for a shape function is typically not influenced by LRD in errors. Finally, we investigate properties of a data-driven bandwidth choice. We show that Averaged Squared Error (ASE) is a good approximation of MISE, however, this is not the case for a cross-validation criterion.