Progress in data-based bandwidth selection for Kernel density estimation

Computational Statistics

Volumn 11, Issue 3, 1996, Pages 337-381

  Jones, M C ^a   Marron, J S ^b   Sheather, S J ^b  

^a OPEN UNIVERSITY   (United Kingdom)

^b University of North Carolina at Chapel Hill   (United States)

Author keywords

Automatic methods; Cross validation; Curve estimation; Functional estimation; Mean integrated squared error; Normal mixture; Oversmoothing; Rates of convergence; Scale estimation; Smoothed boot strapping; Smoothing parameter

Indexed keywords

EID: 0030517778     PISSN: 09434062     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (87)

1. Abramson, I.S. (1982), On bandwidth variation in kernel estimates - a square root law. Annals of Statistics, 10, 1217-1223.
2. Aldershof, B. (1991), Estimation of integrated squared density derivatives. Mimeo Series #2053, Institute of Statistics, University of North Carolina.
3. Berlinet, A. and Devroye, L. (1994), A comparison of kernel density estimates. Publications de l'Institut de Statistique de l'Université de Paris, 38, 3-59.
4. Bowman, A.W. (1984), An alternative method of cross-validation for the smoothing of density estimates. Biometrika, 71, 353-360.
5. Bowman, A.W. (1985), A comparative study of some kernel-based nonparametric density estimators. Journal of Statistical Computation and Simulation, 21, 313-327.
6. Cao, R. (1993), Bootstrapping the mean integrated squared error. Journal of Multivariate Analysis, 45, 137-160.
7. Cao, R., Cuevas, A. and González-Manteiga, W. (1994), A comparative study of several smoothing methods in density estimation. Computational Statistics and Data Analysis, 17, 153-176.
8. Cao-Abad, R. (1990), Aplicaciones y nuevos resultados del método bootstrap en la estimación no paramétrica de curvas. Ph.D. thesis, University of Santiago de Compostela.
9. Cheng, M.-Y. (1994), Robust estimators of integrated density derivative products, to appear.
10. Chiu, S.-T. (1990), On the asymptotic distributions of bandwidth estimates. Annals of Statistics, 18, 1696-1711.
11. Chiu, S.-T. (1991a), Bandwidth selection for kernel density estimation. Annals of Statistics, 19, 1883-1905.
12. Chiu, S.-T. (1991b), The effect of discretization error on bandwidth selection for kernel density estimation. Biometrika, 78, 436-441.
13. Chiu, S.-T. (1991c), Some stabilized bandwidth selectors for nonparametric regression. Annals of Statistics, 19, 1528-1546.
14. Chiu, S.-T. (1992), An automatic bandwidth selector for kernel density estimate. Biometrika, 79, 771-782.
15. Deheuvels, P. (1977), Estimation non paramétrique de la densité par histogrammes généralisés. Revue Statistique Appliqué, 25, 5-42.
16. Devroye, L. (1989), The double kernel method in density estimation. Annales de l'Institut Henri Poincaré, 25, 533-580.
17. 1View, Wiley, New York.
18. Donoho, D.L. (1988), One-sided inference about functionals of a density. Annals of Statistics, 16, 1390-1420.
19. Engel, J., Herrmann, E. and Gasser, T. (1994), An iterative bandwidth selector for kernel estimation of densities and their derivatives. Journal of Nonparametric Statistics, 4, 21-34.
20. Falk, M. (1992), Bootstrap optimal bandwidth selection for kernel density estimation. Journal of Statistical Planning and Inference, 30, 13-22.
21. Fan, J., Hall, P., Martin, M. and Patil, P. (1993), On local smoothing of non-parametric curve estimators. Research Report CMA-SR23-93, Centre for Mathematics and its Applications, Australian National University.
22. Fan, J. and Marron, J.S. (1992), Best possible constant for bandwidth selection. Annals of Statistics, 20, 2057-2070.
23. Faraway, J.J. and Jhun, M. (1990), Bootstrap choice of bandwidth for density estimation. Journal of the American Statistical Association, 85, 1119-1122.
24. Feluch, W. and Koronacki, J. (1992), A note on modified cross-validation in density estimation. Computational Statistics and Data Analysis, 15, 143-151.
25. Gasser, T., Kneip, A. and Köhler, W. (1991), A flexible and fast method for automatic smoothing. Journal of the American Statistical Association, 86, 643-652.
26. Goldstein, L. and Messer, K. (1992), Optimal plug-in estimators for non-parametric functional estimation. Annals of Statistics, 20, 1306-1328.
27. Grund, B., Hall, P. and Marron, J.S. (1994), Loss and risk in smoothing parameter selection. Journal of Nonparametric Statistics, 4, 107-132.
28. Györfi, L., Härdle, W., Sarda, P. and Vieu, P. (1989), Nonparametric Curve Estimation from Time Series, Springer-Verlag, Berlin.
29. Hall, P. (1980), Objective methods for the estimation of window size in the nonparametric estimation of a density, unpublished manuscript.
30. Hall, P. (1987), On Kullback-Liebler loss and density estimation. Annals of Statistics, 15, 1491-1519.
31. Hall, P. (1993), On plug-in rules for local smoothing of density estimators. Annals of Statistics, 21, 694-710.
32. Hall, P., DiCiccio, T.J. and Romano, J.P. (1989) On smoothing and the bootstrap. Annals of Statistics, 17, 692-704.
33. Hall, P. and Johnstone, I.M. (1992), Empiricalfunctionals and efficient smoothing parameter selection (with discussion). Journal of the Royal Statistical Society, Series B, 54, 475-530.
34. Hall, P. and Marron, J.S. (1987a), Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation. Probability Theory and Related Fields, 74, 567-581.
35. Hall, P. and Marron, J.S. (1987b), On the amount of noise inherent in band-width selection for a kernel density estimator. Annals of Statistics, 15, 163-181.
36. Hall, P. and Marron, J.S. (1987c), Estimation of integrated squared density derivatives. Statistics and Probability Letters, 6, 109-115.
37. Hall, P. and Marron, J.S. (1991a), Lower bounds for bandwidth selection in density estimation. Probability Theory and Related Fields, 90, 149-173.
38. Hall, P. and Marron, J.S. (1991b), Local minima in cross-validation functions. Journal of the Royal Statistical Society, Series B, 53, 245-252.
39. Hall, P., Marron, J.S. and Park, B.-U. (1992), Smoothed cross-validation. Probability Theory and Related Fields, 92, 1-20.
40. Hall, P., Sheather, S.J., Jones, M.C. and Marron, J.S. (1991), On optimal data-based bandwidth selection in kernel density estimation. Biometrika, 78, 263-269.
41. 1 distance in nonparametric density estimation. Journal of Multivariate Analysis, 26, 59-88.
42. Hall, P. and Wand, M.P. (1988b), On the minimization of absolute distance in kernel density estimation. Statistics and Probability Letters, 6, 311-314.
43. Härdle, W., Hall, P. and Marron, J.S. (1988), How far are automatically chosen regression smoothing parameters from their optimum? (with comments). Journal of the American Statistical Association, 83, 86-101.
44. Härdle, W., Hall, P. and Marron, J.S. (1992), Regression smoothing parameters that are not far from their optimum. Journal of the American Statistical Association, 87, 227-233.
45. Izenman, A.J. and Sommer, C.J. (1988), Philatelic mixtures and multimodal densities. Journal of the American Statistical Association, 83, 941-953.
46. Janssen, P., Marron, J.S., Veraverbeke, N. and Sarle, W. (1995), Scale measures for bandwidth selection. Journal of Nonparametric Statistics, 5, 359-380.
47. Jones, M.C. (1991a), The roles of ISE and MISE in density estimation. Statistics and Probability Letters, 12, 51-56.
48. Jones, M.C. (1991b), Kernel density estimation for length biased data. Biometrika, 78, 511-519.
49. Jones, M.C. (1992), Potential for automatic bandwidth choice in variations on kernel density estimation. Statistics and Probability Letters, 13, 351-356.
50. Jones, M.C. and Kappenman, R.F. (1992), On a class of kernel density estimate bandwidth selectors. Scandinavian Journal of Statistics, 19, 337-350.
51. Jones, M.C., Marron, J.S. and Park, B.-U. (1991), A simple root n bandwidth selector. Annals of Statistics, 19, 1919-1932.
52. Jones, M.C., Marron, J.S. and Sheather, S.J. (1996), A brief survey of band-width selection for density estimation. Journal of the American Statistical Association, to appear.
53. Jones, M.C. and Sheather, S.J. (1991), Using non-stochastic terms to advantage in estimating integrated squared density derivatives. Statistics and Probability Letters, 11, 511-514.
54. Kim, W.C., Park, B.U. and Marron, J.S. (1994), Asymptotically best band-width selectors in kernel density estimation. Statistics and Probability Letters, 19, 119-127.
55. Marron, J.S. (1986), Will the art of smoothing ever become a science? Contemporary Mathematics, 59, 169-178.
56. Marron, J.S. (1989), Automatic smoothing parameter selection: a survey. Empirical Economics, 13, 187-208.
57. Marron, J.S. (1991), Root n bandwidth selection. In Nonparametric Functional Estimation and Related Topics, ed. G.G. Roussas, Klüwer, Dordrecht, pp. 251-260.
58. Marron, J.S. (1992), Bootstrap bandwidth selection. In Exploring the Limits of the Bootstrap, eds. R. LePage, and L. Billard, Wiley, New York, pp. 249-262.
59. Marron, J.S. and Tsybakov, A.B. (1995), Visual error criteria for qualitative smoothing. Journal of the American Statistical Association, 90, 499-507.
60. Marron, J.S. and Wand, M.P. (1992), Exact mean integrated squared error. Annals of Statistics, 20, 712-736.
61. Müller, H.-G. (1985), Empirical bandwidth choice for nonparametric kernel regression by means of pilot estimators. Statistics and Decisions, Supplementary Issue #2, 193-206.
62. Park, B.-U. (1989), On the plug-in bandwidth selectors in kernel density estimation. Journal of the Korean Statistical Society, 18, 107-117.
63. Park, B.-U. and Cho, S. (1991), Estimation of integrated squared spectral density derivatives. Statistics and Probability Letters, 12, 65-72.
64. Park, B.-U. and Marron, J.S. (1990), Comparison of data-driven bandwidth selectors. Journal of the American Statistical Association, 85, 66-72.
65. Park, B.-U. and Marron, J.S. (1992), On the use of pilot estimators in band-width selection. Journal of Nonparametric Statistics, 1, 231-240.
66. Park, B.-U. and Turlach, B.A. (1992), Practical performance of several data driven bandwidth selectors (with discussion). Computational Statistics, 7, 251-270.
67. Rudemo, M. (1982), Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics, 9, 65-78.
68. Ruppert, D., Sheather, S.J. and Wand, M.P. (1995), An effective bandwidth selector for local least squares regression. Journal of the American Statistical Association, 90, 1257-1270.
69. Sain, S.R., Baggerly, K.A. and Scott, D.W. (1994), Cross-validation of multivariate densities. Journal of the American Statistical Association, 89, 807-817.
70. Scott, D.W., Tapia, R.A. and Thompson, J.R. (1977), Kernel density estimation revisited. Nonlinear Analysis, Theory, Methods and Applications, 1, 339-372.
71. Scott, D.W. and Terrell, G.R. (1987), Biased and unbiased cross-validation in density estimation. Journal of the American Statistical Association, 82, 1131-1146.
72. Sheather, S.J. (1986), An improved data-based algorithm for choosing the window width when estimating the density at a point. Computational Statistics and Data Analysis, 4, 61-65.
73. Sheather, S.J. (1992), The performance of six popular bandwidth selection methods on some real data sets (with discussion). Computational Statistics, 7, 225-250.
74. Sheather, S.J. and Jones, M.C. (1991), A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society, Series B, 53, 683-690.
75. Silverman, B.W. (1986), Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
76. Stute, W. (1992), Modified cross-validation in density estimation. Journal of Statistical Planning and Inference, 30, 293-305.
77. Taylor, C.C. (1989), Bootstrap choice of the smoothing parameter in kernel density estimation. Biometrika, 76, 705-712.
78. Terrell, G.R. (1990), The maximal smoothing principle in density estimation. Journal of the American Statistical Association, 85, 470-477.
79. Terrell, G.R. and Scott, D.W. (1985), Oversmoothed nonparametric density estimates. Journal of the American Statistical Association, 80, 209-214.
80. Thombs, L.A. and Sheather, S.J. (1992), Local bandwidth selection for density estimation. In Proceedings of the 22nd Symposium on the Interface, Springer-Verlag, New York, pp. 111-116.
81. van Es, A.J. (1992), Asymptotics for least squares cross-validation band-widths in nonsmooth cases. Annals of Statistics, 20, 1647-1657.
82. van Es, A.J. and Hoogstrate, A. (1994), How much do plug-in bandwidth selectors adapt to non-smoothness? Research Report RM/0/94-051, Maastricht Research School of Economics of Technology and Organizations.
83. Wand, M.P., and Jones, M.C. (1994), Multivariate plug-in bandwidth selection. Computational Statistics, 9, 97-116.
84. Wand, M.P., Marron, J.S. and Ruppert, D. (1991), Transformations in density estimation (with comments). Journal of the American Statistical Association, 86, 343-361.
85. Weisberg, S. (1980), Applied Linear Regression, Wiley, New York.
86. Woodroofe, M. (1970), On choosing a delta-sequence. Annals of Mathematical Statistics, 41, 1665-1671.
87. Young, G.A. (1990), Alternative smoothed bootstraps. Journal of the Royal Statistical Society, Series B, 52, 477-484.