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# Empirical Functionals and Efficient Smoothing Parameter Selection

Peter Hall and Iain Johnstone
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 54, No. 2 (1992), pp. 475-530
Stable URL: http://www.jstor.org/stable/2346138
Page Count: 56
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## Abstract

A striking feature of curve estimation is that the smoothing parameter $\hat h_0$, which minimizes the squared error of a kernel or smoothing spline estimator, is very difficult to estimate. This is manifest both in slow rates of convergence and in high variability of standard methods such as cross-validation. We quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of $\hat h_0$ that attain the bounds. The efficient estimators are substantially less variable than cross-validation (and other current procedures) and simulations suggest that they may offer improvements at moderate sample sizes, at least in terms of minimizing the squared error. The key is a stochastic decomposition of the empirical functional $\hat h_0$ in terms of a smooth quadratic functional of the unknown curve. Examples include the estimation of densities, regression functions and continuous signals in Gaussian white noise.

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