Biased and Unbiased Cross-Validation in Density Estimation
David Scott,George R. Terrell +1 more
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In this article, biased cross-validation criteria for selection of smoothing parameters for kernel and histogram density estimators, closely related to one investigated in Scott and Factor (1981), were introduced.Abstract:
Nonparametric density estimation requires the specification of smoothing parameters. The demands of statistical objectivity make it highly desirable to base the choice on properties of the data set. In this article we introduce some biased cross-validation criteria for selection of smoothing parameters for kernel and histogram density estimators, closely related to one investigated in Scott and Factor (1981). These criteria are obtained by estimating L 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared L 2 error or mean integrated squared error. These criteria are roughly the analog of Wahba's (1981) generalized cross-validation procedure for orthogonal series density estimators. We present the relationship of the biased cross-validation procedure to the least squares cross-validation procedure, which provides unbiased estimates of the mean integrated squared error. Both methods are shown to be based on U statistics. We compare the two metho...read more
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TL;DR: In this article, the problem of choosing a bandwidth parameter for nonparametric regression is studied and the relationship of this estimate to a kernel estimate is discussed, based on an unbiased estimate of mean square error, which is shown to be asymptotically optimal.