Nonparametric Estimation in a Model with a Trend

Let Yt be a stochastic process taking values in R and defined by the model Yt=azt+φ(Xt)+ εt where {zt} is a deterministic sequence, {Xt} is strictly stationary and strongly mixing, and {εt} is i.i.d. We study asymptotic properties of nonparametric estimators of density and regression with rates of convergence, and their behavior on estimation when φ(ċ) is polynomial. It is shown that the estimator of the coefficients of φ(ċ) constructed from the nonparametric estimators of regression is consistent when the deterministic {zt} converges in Cesàro mean, and may be inconsistent when {zt} is periodic under some ordinary conditions.