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On plug-in rules for local smoothing of density estimators. (English) Zbl 0779.62035
Summary: Optimal local smoothing of a curve estimator requires knowledge of various derivatives of the curve in the neighbourhood of the point at which estimation is being conducted. One empirical approach to selecting the amount of smoothing is to employ pilot estimators to approximate those derivatives, and substitute the approximate values into an analytical formula for the desired local bandwidth. We study how bandwidth choice for the pilot estimators affects the performance of the final estimator.
Our conclusions are rather curious. Depending on circumstance, the pilot estimators should be substantially oversmoothed or undersmoothed, relative to the amount of smoothing that would be optimal if they were to be employed themselves for point estimation. Occasionally, the optimal amount of undersmoothing is so extreme as to render the pilot estimators inconsistent. Here, the resulting local bandwidth is asymptotically random; it is not asymptotic to a sequence of constants.

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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