Bandwidth Selection for Local Density Estimators

The performance of a kernel density estimator depends crucially on the size of its smoothing bandwidth. A data-driven bandwidth selector for density estimation at a point is proposed in this paper. The technique is based upon minimization of a smoothed bootstrap estimate of the mean squared error of the density estimate. The resultant bandwidth selector can perform better than earlier methods based upon asymptotic approximations to the mean squared error. This is because large sample theory does not always provide a useful approximation to finite sample behaviour in local density estimation.

[1]  Simon J. Sheather,et al.  Local Bandwidth Selection for Density Estimation , 1992 .

[2]  P. Diggle,et al.  Monte Carlo Methods of Inference for Implicit Statistical Models , 1984 .

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

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

[5]  Martin L. Hazelton,et al.  Improved Monte Carlo inference for models with additive error , 1995 .

[6]  Peter Hall,et al.  A local cross-validation algorithm , 1989 .

[7]  D. W. Scott,et al.  Multivariate Density Estimation, Theory, Practice and Visualization , 1992 .

[8]  B. Efron.  Bootstrap Methods: Another Look at the Jackknife , 1979 .

[9]  W. R. Schucany.  Locally optimal window widths for kernel density estimation with large samples , 1989 .

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

[11]  M. C. Jones,et al.  A Brief Survey of Bandwidth Selection for Density Estimation , 1996 .

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

[13]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

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

[15]  David W. Scott,et al.  Feasibility of multivariate density estimates , 1991 .

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

[17]  Jerome Sacks,et al.  ASYMPTOTICALLY OPTIMUM KERNELS FOR DENSITY ESTIMATION AT A POINT , 1981 .

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

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

[20]  A. A. Chilingarian,et al.  A bootstrap method of distribution mixture proportion determination , 1990, Pattern Recognit. Lett..

[21]  J. Stephen Marron,et al.  Bootstrap bandwidth selection , 1990 .

[22]  Jan Mielniczuk,et al.  Local data-driven bandwidth choice for density estimation , 1989 .

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

[24]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .

[25]  S. Sheather.  A data-based algorithm for choosing the window width when estimating the density at a point , 1983 .

[26]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[27]  A. Izenman.  Recent Developments in Nonparametric Density Estimation , 1991 .