EMPIRICAL FUNCTIONALS AND EFFICIENT SMOOTHING PARAMETER SELECTION

A striking feature of curve estimation is that the smoothing parameter h0, which minimizes the squared error of a kernel or smoothing spline estimator, is very difficult to estimate. This is manifest both in sl ow rates of convergence and in high variability of standard methods su ch as cross-validation. We quantify this difficulty by describing nonp arametric information bounds and exhibit asymptotically efficient esti mators of h0 that attain the bounds. The efficient estimators are subs tantially less variable than cross-validation (and other current proce dures) and simulations suggest that they may offer improvements at mod erate sample sizes, at least in terms of minimizing the squared error. The key is a stochastic decomposition of the empirical functional h0 in terms of a smooth quadratic functional of the unknown curve. Exampl es include the estimation of densities, regression functions and conti nuous signals in Gaussian white noise.