Smoothed cross-validation. (English) Zbl 0742.62042

Summary: 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.


62G07 Density estimation
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference


Full Text: DOI


[1] Bickel, P., Ritov, Y.: Estimating integrated squared density derivatives. Sankhy? Ser. A.50, 381-393 (1988) · Zbl 0676.62037
[2] Bowman, A. W.: An alternative method of cross-validation for the smoothing of density estimates. Biometrika71, 353-360 (1984) · doi:10.1093/biomet/71.2.353
[3] Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Probab.1, 19-42 (1973) · Zbl 0301.60035 · doi:10.1214/aop/1176997023
[4] Burman, P.: A data dependent approach to density estimation, Z. Wahrscheinlichkeits theor. Verw. Geb.69, 609-628 (1985) · Zbl 0586.62044 · doi:10.1007/BF00532670
[5] Chiu, S.-T.: Bandwidth selection for kernel density estimation. Ann. Stat. to appear (1991) · Zbl 0749.62022
[6] Devroye, L., Gy?rfi, L.: Nonparametric density estimation: TheL 1 View. New York: Wiley 1984
[7] Diggle, P.J.: Statistical analysis of point patterns London: Academic Press 1983 · Zbl 0559.62088
[8] Diggle, P.J.: A kernel method for smoothing point process data. Appl. Stat.34, 138-147 (1985) · Zbl 0584.62140 · doi:10.2307/2347366
[9] Diggle, P.J., Marron, J.S.: Equivalence of smoothing parameter selectors in density and intensity estimation. J. Am. Stat. Assoc.83, 793-800 (1988) · Zbl 0662.62036 · doi:10.2307/2289308
[10] Fubank, R.L.: Spline smoothing and nonparametric regression. New York: Dekker 1988
[11] Faraway, J.J., Jhun, M.: Bootstrap choice of bandwidth for density estimation. J. Am. Stat. Assoc.85, 1119-1122 (1990) · doi:10.2307/2289609
[12] Gasser, T., M?ller, H-G., Mammitzsch, V.: Kernels for nonparametric curve estimation. J. R. Stat. Soc., Ser.B47, 238-252 (1985) · Zbl 0574.62042
[13] H?rdle, W.: Applied nonparametric regression. Econometrics Society Monograph Series, No. 19, Cambridge: Cambridge University Press 1989
[14] H?rdle, W., Hall, P., Marron, J.S.: How far are automatically chosen regression smoothers from their optimum? (with discussion). J. Am. Stat. Assoc.83, 86-95 (1988) · Zbl 0644.62048 · doi:10.2307/2288922
[15] Hall, P.: Objective methods for the estimation of window size in the nonparametric estimation of a density (unpublished manuscript, 1980)
[16] Hall, P.: Large sample optimality of least squares cross-validation in density estimation. Ann. Stat.11, 1156-1174 (1983) · Zbl 0599.62051
[17] Hall, P.: Central limit theorem for integrated squared error of multivariate density estimators. J. Multivariate Anal.14, 1-16 (1984) · Zbl 0528.62028 · doi:10.1016/0047-259X(84)90044-7
[18] Hall, P., Marron, J.S.: Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation. Probab. Th. Rel. Fields74, 567-581 (1987a) · Zbl 0588.62052 · doi:10.1007/BF00363516
[19] Hall, P., Marron, J.S.: On the amount of noise inherent in bandwidth selection for a kernel density estimator. Ann. Stat. 15, 163-181 (1987b) · Zbl 0667.62022 · doi:10.1214/aos/1176350259
[20] Hall, P., Marron, J.S.: Estimation of integrated squared density derivatives. Stat. Probab. Lett.6, 109-115 (1987c) · Zbl 0628.62029 · doi:10.1016/0167-7152(87)90083-6
[21] Hall, P., Marron, J.S.: Lower bounds for bandwidth selection in density estimation (unpublished manuscript, 1989) · Zbl 0742.62041
[22] Hall, P., Sheather, S., Jones, M.C., Marron, J.S.: On optimal data-based bandwidth selection in kernel density estimation. (unpublished manuscript, 1989) · Zbl 0733.62045
[23] Marron, J.S.: Automatic smoothing parameter selection: A survey. Emp. Econ.13, 187-208 (1988) · doi:10.1007/BF01972448
[24] M?ller, H.G.: Empirical bandwidth choice for nonparametric kernel regression by means of pilot estimators. Stat. Decis.2, [Suppl] 193-206 (1985)
[25] M?ller, H.G.: Nonparametric analysis of longitudinal data. Berlin Heidelberg New York: Springer 1988
[26] Park, B.U., Marron, J.S.: Comparison of data-driven bandwidth selectors. J. Am. Stat. Assoc.85, 66-72 (1990) · doi:10.2307/2289526
[27] Ripley, B.D.: Spatial statistics New York: Wiley 1981 · Zbl 0583.62087
[28] Rudemo, M.: Empirical choice of histograms and kernel density estimators. Scand. J. Stat.9, 65-78 (1982) · Zbl 0501.62028
[29] Scott, D.W.: Averaged shifted histograms: effective nonparametric density estimation in several dimensions. Ann. Stat.4, 1024-1040 (1985) · Zbl 0589.62022 · doi:10.1214/aos/1176349654
[30] Scott, D.W., Factor, L.E.: Monte Carlo study of three data-based nonparametric density estimators. J. Am. Stat. Assoc.76, 9-15 (1981) · Zbl 0465.62036 · doi:10.2307/2287033
[31] Scott, D.W., Terrell, G.R.: Biased and unbiased cross-validation in density estimation. J. Am. Stat. Assoc.82, 1131-1146 (1987) · Zbl 0648.62037 · doi:10.2307/2289391
[32] Scott, D.W., Tapia, R.A., Thompson, J.W.: Kernel density estimation revisited. J. Nonlinear Anal., Theor. Methods Appl.1, 339-372 (1977) · Zbl 0363.62030 · doi:10.1016/S0362-546X(97)90003-1
[33] Sheather, S.J.: An improved data-based algorithm for choosing the window width when estimating the density at a point. Comput. Stat. Data Anal.4, 61-65 (1986) · doi:10.1016/0167-9473(86)90026-5
[34] Silverman, B.W.: Density estimation for statistics and data analysis. New York: Chapman and Hall 1986 · Zbl 0617.62042
[35] Staniswallis, J.G.: Local bandwidth selection for kernel estimates. J. Am. Stat. Assoc.84, 284-288 (1987) · Zbl 0676.62039 · doi:10.2307/2289875
[36] Stone, C.J.: An asymptotically optimal window selection rule for kernel density estimates. Ann. Stat.12, 1285-1297 (1984) · Zbl 0599.62052 · doi:10.1214/aos/1176346792
[37] Taylor, C.C.: Bootstrap choice of the smoothing parameter in kernel density estimation Biometrika76, 705-712 (1989) · Zbl 0678.62042 · doi:10.1093/biomet/76.4.705
[38] Woodroofe, M.: On choosing a delta sequence. Ann. Math. Stat.41, 1665-1671 (1970) · Zbl 0229.62022 · doi:10.1214/aoms/1177696810
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