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September, 1985 Spline Smoothing and Optimal Rates of Convergence in Nonparametric Regression Models
Paul Speckman
Ann. Statist. 13(3): 970-983 (September, 1985). DOI: 10.1214/aos/1176349650


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.


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Paul Speckman. "Spline Smoothing and Optimal Rates of Convergence in Nonparametric Regression Models." Ann. Statist. 13 (3) 970 - 983, September, 1985.


Published: September, 1985
First available in Project Euclid: 12 April 2007

zbMATH: 0585.62074
MathSciNet: MR803752
Digital Object Identifier: 10.1214/aos/1176349650

Primary: 62J05
Secondary: 41A15, 62G35

Rights: Copyright © 1985 Institute of Mathematical Statistics


Vol.13 • No. 3 • September, 1985
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