Adaptive estimation in partly linear regression models

Consider the regression model y _i = x _iβ + g(t _i) + e _i for i = 1, 2, h., n. Here g(·) is an unknown function, β is a parameter to be estimated, and e _i are random errors. Based on g(·) estimated by kernel type estimator for the case where (x _i, t _i) are i. i. d. design points, the adaptive estimator of β is investigated, and some results about the asymptotically optimal convergence rates of the estimates are also obtained. In the meantime, the family of nonparametric estimates of g(·) including the known kernel and nearest neighbor estimates is proposed. Based on the nonparametric estimate for the case that (x _i, t _i) are known and nonrandom, the asymptotic normality of least squares estimator of β is proved.