# 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.

**Keywords**:

**JEL-Classification-Number**:

**Creation-Date**: June 1986

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