Bias corrected bootstrap bandwidth selection

Current bandwidth selectors for kernel density estimation that are asymptotically optimal often prove less promising under more moderate sample sizes. The point of this paper is to derive a class of bandwidth selectors that attain optimal root-n convergence while still showing good results under small and moderate sample sizes. This is achieved by minimizing bias corrected smoothed bootstrap estimates of the mean integrated squared error. The degree of bias correction determines the rate of relative convergence of the resulting bandwidth. The bias correction targets finite sample bias rather than asymptotically leading terms, resulting in substantial improvements under smaller sample sizes. In a simulation study, the new methods are compared to several of the currently most popular bandwidth selectors, including plug-in, cross-validation and other bootstrap rules. Practical implementation, and the choice of the oversmoothing bandwidth are discussed.

[1]  J. Simonoff.  Multivariate Density Estimation , 1996 .

[2]  Brian Kent Aldershof,et al.  Estimation of integrated squared density derivatives , 1991 .

[3]  James Stephen Marron,et al.  Comparison of data-driven bandwith selectors , 1988 .

[4]  J. Faraway,et al.  Bootstrap choice of bandwidth for density estimation , 1990 .

[5]  Loss and risk in smoothing parameter selection , 1994 .

[6]  M. C. Jones,et al.  A reliable data-based bandwidth selection method for kernel density estimation , 1991 .

[7]  M. Rudemo.  Empirical Choice of Histograms and Kernel Density Estimators , 1982 .

[8]  G. A. Young,et al.  The bootstrap: To smooth or not to smooth? , 1987 .

[9]  M. C. Jones.  The roles of ISE and MISE in density estimation , 1991 .

[10]  A. Bowman.  An alternative method of cross-validation for the smoothing of density estimates , 1984 .

[11]  J. Marron,et al.  Progress in data-based bandwidth selection for kernel density estimation , 1996 .

[12]  Charles C. Taylor,et al.  Bootstrap choice of the smoothing parameter in kernel density estimation , 1989 .

[13]  P. Spreij.  Probability and Measure , 1996 .

[14]  J. Marron,et al.  Smoothed cross-validation , 1992 .

[15]  Peter Hall,et al.  Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems , 1990 .

[16]  Patrick Billingsley,et al.  Probability and Measure. , 1986 .

[17]  James Stephen Marron,et al.  A simple root n bandwidth selector , 1991 .

[18]  James Stephen Marron,et al.  Lower bounds for bandwidth selection in density estimation , 1991 .

[19]  M. C. Jones,et al.  On a class of kernel density estimate bandwidth selectors , 1991 .

[20]  M. Wand,et al.  EXACT MEAN INTEGRATED SQUARED ERROR , 1992 .

[21]  J. Marron,et al.  Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation , 1987 .

[22]  Richard A. Becker,et al.  The New S Language , 1989 .

[23]  B. Turlach,et al.  Rejoinder to ``Practical performance of several data driven bandwidth selectors" , 1992 .