Spline Smoothing and Optimal Rates of Convergence in Nonparametric Regression Models

Abstract

Linear estimation is considered in nonparametric regression models of the form $Y_i = f(x_i) + \varepsilon_i, x_i \in (a, b)$, where the zero mean errors are uncorrelated with common variance $\sigma^2$ and the response function $f$ is assumed only to have a bounded square integrable $q$th derivative. The linear estimator which minimizes the maximum mean squared error summed over the observation points is derived, and the exact minimax rate of convergence is obtained. For practical problems where bounds on $\|f^{(q)}\|^2$ and $\sigma^2$ may be unknown, generalized cross-validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality.