Asymptotic Equivalence of Nonparametric Autoregression and Nonparametric Regression

where (εi)i=1,...,n are i.i.d. random variables. The unknown autoregression function f is then the target of statistical inference and the development of efficient estimators is a natural task for theoretically oriented statisticians. On the one hand, it has been recognized for a long time that commonly used estimators in model (1) have the same asymptotic behavior as corresponding estimators in nonparametric regression. A result of Robinson [26] concerns the pointwise equivalence of nonparametric kernel estimators and Neumann and Kreiss [22] extended this equivalence to the global behavior of nonparametric estimators. On the other hand, despite these well-known similarities between estimators, there is still a certain discrepancy in the current state of available theory in both contexts. While there is a very well developed asymptotic theory for optimal estimation in nonparametric regression, even up to the level of exact asymptotics (see, e.g., [13] or [24], for an overview), there is considerably less theory available in the case of nonparametric autoregression.