SFB 303 Discussion Paper No. A - 201

Author: Hall, Peter, and J. S. Marron
Title: Local Minima in Cross-Validation Functions
Abstract: The method of least squares cross-validation for choosing the bandwidth of a kernel density estimator has been the object of considerable research, both through theoretical analysis and also simulation studies. The method involves the minimization of a certain function of the bandwidth. One of the less attractive features of this method, which has been observed in simulation studies but has not previously been understood theoretically, is that rather often the cross-validation function has multiple local minima. The theoretical results of this paper provide an explanation and quantification of these local minima through modeling the cross-validation function as a Gaussian stochastic process. Asymptotic analysis reveals that the expected number of local minima depends on the underlying density through a fairly simple functional, but dependence on the kernel function is much more complicated. An interesting feature is that the expected number of minima increases with a quantity which can be interpreted as measuring the "simplicity" of the density. A simulation study explores the extent to which the asymptotic analysis describes the actual situation.
Keywords: Bandwidth selection, Cross-validation, Kernel density estimation, Local minima, Smoothing parameter selection
Creation-Date: September 1988
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