Abstract: | Let $\{$(X$\sb{\rm t}$,Y$\sb{\rm t}$): t $\in$ N$\}$ be a strictly stationary process with X$\sb{\rm t}$ being R$\sp{\rm d}$-valued and Y$\sb{\rm t}$ being real valued. Consider the problem of estimating the conditional expectation function, m(x) = E(Y$\sb{\rm t}\vert$ X$\sb{\rm t}$ = x), using (X$\sb1,$Y$\sb1$),$\...$ (X$\sb{\rm n}$,Y$\sb{\rm n}$). (For example, suppose Z$\sb{\rm t}$, t = 0, $\pm$1, $\pm$2,.. is a real valued stationary time series and p is a positive integer. Set X$\sb{\rm t}$ = (Z$\sb{\rm t+1},\...$,Z$\sb{\rm t+d}$) and Y$\sb{\rm t}$ = Z$\sb{\rm t+d+p}$. Then (X$\sb{\rm t}$,Y$\sb{\rm t}$), t = 0, $\pm$1,.. is a stationary time series and m(x) = E(Z$\sb{\rm d+p}\vert$Z$\sb1,\...$Z$\sb{\rm d}$).) We consider kernel estimators of m(x). Recently, convergence properties of the kernel estimator have been developed under certain dependence structures for the process (X$\sb{\rm t}$,Y$\sb{\rm t}$). One of the crucial points in applying a kernel estimator is the choice of bandwidth. The main purpose of this work is to establish asymptotic optimality for a bandwidth selection rule under dependence which can be interpreted in terms of cross validation. In addition, some moment bounds for dependent variables will be established, which give more flexible bounds than existing ones. |