Probability Theory and Related Fields

, Volume 92, Issue 1, pp 1-20

First online:

Smoothed cross-validation

  • Peter HallAffiliated withAustralian National UniversitySeoul National University
  • , J. S. MarronAffiliated withAustralian National UniversitySeoul National University
  • , Byeong U. ParkAffiliated withAustralian National UniversitySeoul National University

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For bandwidth selection of a kernel density estimator, a generalization of the widely studied least squares cross-validation method is considered. The essential idea is to do a particular type of “presmoothing” of the data. This is seen to be essentially the same as using the smoothed bootstrap estimate of the mean integrated squared error. Analysis reveals that a rather large amount of presmoothing yields excellent asymptotic performance. The rate of convergence to the optimum is known to be best possible under a wide range of smoothness conditions. The method is more appealing than other selectors with this property, because its motivation is not heavily dependent on precise asymptotic analysis, and because its form is simple and intuitive. Theory is also given for choice of the amount of presmoothing, and this is used to derive a data-based method for this choice.