A universally acceptable smoothing factor for kernel density estimates

Annals of Statistics

Volumn 24, Issue 6, 1996, Pages 2499-2512

  Devroye, Luc ^a   Lugosi, Gábor ^a  

^a NONE

Author keywords

Asymptotic optimality; Convergence; Density estimation; Kernel estimate; Minimum distance estimate; Smoothing factor

Indexed keywords

EID: 0030335054     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1032181164     Document Type: Article

References (41)

1. AKAIKE, H. (1954). An approximation to the density function. Ann. Inst. Statist. Math. 6 127-132.
2. ASSOUAD, P. (1983). Deux remarques sur l'estimation. C. R. Acad. Sci. Paris Sér. I Math. 296 1021-1024.
3. BERLINET, A. and DEVROYE, L. (1994). A comparison of kernel density estimates. Publ. Inst. Statist. Univ. Paris 38 3-59.
4. BIRGÉ, L. (1985). Non-asymptotic minimax risk for Hellinger balls. Probab. Math. Statist. 5 21-29.
5. BIRGÉ, L. (1986). On estimating a density using Hellinger distance and some other strange facts. Probab. Theory Related Fields 71 271-291.
6. BIRGÉ, L. (1987a). On the risk of histograms for estimating decreasing densities. Ann. Statist. 15 1013-1022.
7. BIRGÉ, L. (1987b). Estimating a density under order restrictions: nonasymptotic minimax risk. Ann. Statist. 15 995-1012.
8. BIRGÉ, L. (1989). The Grenander estimator: a nonasymptotic approach. Ann. Statist. 17 1532-1549.
9. BOWMAN, A. W. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71 353-360.
10. BRETAGNOLLE, J. and HUBER, C. (1979). Estimation des densités: risque mimimax. Z. Wahrsch. Verw. Gebiete 47 119-137.
11. CAO, R., CUEVAS, A. and GONZÁLEZ-MANTEIGA, W. (1994). A comparative study of several smoothing methods in density estimation. Comput. Statist. Data Anal. 17 153-176.
12. DEHEUVELS, P. (1977a). Estimation non paramétrique de la densité par histogrammes generalisés. Rev. Statist. Appl. 25 5-42.
13. DEHEUVELS, P. (1977b). Estimation nonparamétrique de la densité par histogrammes generalisés. Publ. Inst. Statist. Univ. Paris 22 1-23.
14. DEHEUVELS, P. and HOMINAL, P. (1980). Estimation automatique de la densité. Rev. Statist. Appl. 28 25-55.
15. 1 for kernel density estimates. Ann. Statist. 11 896-904.
16. DEVROYE, L. (1987). A Course in Density Estimation. Birkhäuser, Boston.
17. DEVROYE, L. (1988). Asymptotic performance bounds for the kernel estimate. Ann. Statist. 16 1162-1179.
18. DEVROYE, L. (1989a). A universal lower bound for the kernel estimate. Statist. Probab. Lett. 8 419-423.
19. DEVROYE, L. (1989b). The double kernel method in density estimation. Ann. Inst. H Poincaré 25 533-580.
20. DEVROYE, L. (1991). Exponential inequalities in nonparametric estimation. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 31-44. Kluwer, Dordrecht.
21. DEVROYE, L. (1992). A note on the usefulness of superkernels in density estimation. Ann. Statist. 20 2037-2056.
22. DEVROYE, L. (1994). On good deterministic smoothing sequences for kernel density estimates. Ann. Statist. 22 886-889.
23. 1 View. Wiley, New York.
24. DEVROYE, L. and PENROD, C. S. (1984). Distribution-free lower bounds in density estimation. Ann. Statist. 12 1250-1262.
25. DONOHO, D. L., KERKYACHARIAN, G. and PICARD, D. (1995). Wavelet shrinkage: asymptotia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301-369.
26. DONOHO, D. L. and LIU, R. C. (1991). Geometrizing rates of convergence. III. Ann. Statist. 19 668-701.
27. HALL, P. (1989). On convergence rates in nonparametric problems. Internat. Statist. Rev. 57 45-58.
28. 1 distance in nonparametric density estimation. J. Multivariate Anal. 26 59-88.
29. HOEFFDING, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30.
30. IBRAGIMOV, I. A. and KHASMINSKII, R. Z. (1982). Estimation of distribution density belonging to a class of entire function. Theory Probab. Appl. 27 551-562.
31. LOW, M. (1992). Non-existence of an adaptive estimator for the value of an unknown probability density. Ann. Statist. 20 598-602.
32. MARRON, J. S. (1988). Automatic smoothing parameter selection: a survey. Empirical Economics 13 187-208.
33. MARRON, J. S. (1989). Automatic smoothing parameter selection: a survey. In Semiparametric and Nonparametric Economics (A. Ullah, ed.) 65-86. Heidelberg.
34. MCDIARMID, C. (1989). On the method of bounded differences. In Surveys in Combinatorics 1989. London Math. Soc. Lecture Notes Ser. 141 148-188. Cambridge Univ. Press.
35. PARK, B.-U. and TURLACH, B. A. (1992). Practical performance of several data driven bandwidth selectors (with discussion). Comput. Statist. 7 251-270.
36. PARZEN, E. (1962). On the estimation of a probability density function and the mode. Ann. Math. Statist. 33 1065-1076.
37. ROSENBLATT, M. (1956). Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 832-837.
38. RUDEMO, M. (1982). Empirical choice of histograms and kernel density estimators. Scand. J. Statist. 9 65-78.
39. STONE, C. J. (1984). An asymptotically optimal window selection rule for kernel density estimates. Ann. Statist. 12 1285-1297.
40. TURLACH, B. A. (1993). Bandwidth selection in kernel density estimation: a review. Technical report, Univ. Catholique de Louvain.
41. YATRACOS, Y. G. (1985). Rates of convergence of minimum distance estimators and Kolmogorov's entropy. Ann. Statist. 13 768-774.