A BERRY-ESSEEN TYPE BOUND OF REGRESSION ESTIMATOR BASED ON LINEAR PROCESS ERRORS

Abstract

Consider the nonparametric regression model <TEX>$Y_{ni}\;=\;g(x_{ni})+{\epsilon}_{ni}$</TEX> (<TEX>$1\;{\leq}\;i\;{\leq}\;n$</TEX>), where g(<TEX>$\cdot$</TEX>) is an unknown regression function, <TEX>$x_{ni}$</TEX> are known fixed design points, and the correlated errors {<TEX>${\epsilon}_{ni}$</TEX>, <TEX>$1\;{\leq}\;i\;{\leq}\;n$</TEX>} have the same distribution as {<TEX>$V_i$</TEX>, <TEX>$1\;{\leq}\;i\;{\leq}\;n$</TEX>}, here <TEX>$V_t\;=\;{\sum}^{\infty}_{j=-{\infty}}\;{\psi}_je_{t-j}$</TEX> with <TEX>${\sum}^{\infty}_{j=-{\infty}}\;|{\psi}_j|$</TEX> < <TEX>$\infty$</TEX> and {<TEX>$e_t$</TEX>} are negatively associated random variables. Under appropriate conditions, we derive a Berry-Esseen type bound for the estimator of g(<TEX>$\cdot$</TEX>). As corollary, by choice of the weights, the Berry-Esseen type bound can attain O(<TEX>$n^{-1/4}({\log}\;n)^{3/4}$</TEX>).