SFB 303 Discussion Paper No. A - 74

Author: Härdle, Wolfgang, Peter Hall, and J. S. Marron
Title: How Far Are Automatically Chosen Regression Smoothing Parameters from their Optimum?
Abstract: The problem of smoothing parameter selection for nonparametric curve estimators is addressed in the specific context of kernel regression estimation. Call the "optimal bandwidth" the minimizer of the average squared error. A number of automatically selected bandwidths which approximate the optimum are considered. How far are the automatically selected bandwidths form the optimum? The answer to this question is studied both theoretically and through simulations. The theoretical results include a central limit theorem which both quantifies the rate of convergence and also gives the asymptotic distribution of the difference. The rate of convergence turns out to be excruciatingly slow. This is not too disappointing because this rate is of the same order as the rate of convergence of the difference between the minimizers of the average squared error and the mean average squared error. In some simulations by John Rice, the selectors considered here performed quite differently form each other. We anticipated that these differences would be reflected in different asymptotic distributions for the various selectors. It is rather surprising that all of the selectors have the same limiting normal distribution. To provide insight into the gap between our theoretical results and the above simulations, we did a further Monte Carlo study. Our simulations support the theoretical results, and suggest that the differences observed by Rice seemed to be principally due to the choice of a very small error standard deviation and the choice of error criterion.
Creation-Date: June 1986 
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