×
Devroye, Luc; Lugosi, Gábor

A universally acceptable smoothing factor for kernel density estimates. (English) Zbl 0867.62024

Ann. Stat. 24, No. 6, 2499-2512 (1996).
Summary: We define a minimum distance estimate of the smoothing factor for kernel density estimates, based on a methodology first developed by Y. G. Yatracos [ibid. 13, 768-774 (1985; Zbl 0576.62057)]. It is shown that if \(f_{nh}\) denotes the kernel density estimate on \(\mathbb{R}^d\) for an i.i.d. sample of size \(n\) drawn from an unknown density \(f\), where \(h\) is the smoothing factor, and if \(f_n\) is the kernel estimate with the same kernel and with the proposed new data-based smoothing factor, then, under a regularity condition on the kernel \(K\), \[ \sup_f \limsup_{n\to\infty} {{{\mathbf E}\int|f_n-f|dx}\over{\inf_{h>0}{\mathbf E}\int|f_{nh}-f|dx}}\leq 3. \] This is the first published smoothing factor that can be proven to have this property.

Cited in 1 Review
Cited in 33 Documents

MSC:

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

Keywords:

kernel; convergence; asymptotic optimality; minimum distance estimate; smoothing factor; kernel density estimates

Citations:

Zbl 0576.62057
Cite Review PDF
Full Text: DOI

References:

[1] Akaike, H. (1954). An approximation to the density function. Ann. Inst. Statist. Math. 6 127-132. · Zbl 0058.12302 · doi:10.1007/BF02900741
[2] Assouad, P. (1983). Deux remarques sur l’estimation. C. R. Acad. Sci. Paris Sér. I Math. 296 1021-1024. · Zbl 0568.62003
[3] Berlinet, A. and Devroy e, L. (1994). A comparison of kernel density estimates. Publ. Inst. Statist. Univ. Paris 38 3-59. · Zbl 0804.62039
[4] Birgé, L. (1985). Non-asy mptotic minimax risk for Hellinger balls. Probab. Math. Statist. 5 21-29. · Zbl 0629.62012
[5] Birgé, L. (1986). On estimating a density using Hellinger distance and some other strange facts. Probab. Theory Related Fields 71 271-291. Birgé, L. (1987a). On the risk of histograms for estimating decreasing densities. Ann. Statist. 15 1013-1022. Birgé, L. (1987b). Estimating a density under order restrictions: nonasy mptotic minimax risk. Ann. Statist. 15 995-1012.
[6] Birgé, L. (1989). The Grenander estimator: a nonasy mptotic approach. Ann. Statist. 17 1532- 1549. · Zbl 0703.62042 · doi:10.1214/aos/1176347380
[7] Bowman, A. W. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71 353-360. JSTOR: · doi:10.1093/biomet/71.2.353
[8] Bretagnolle, J. and Huber, C. (1979). Estimation des densités: risque mimimax. Z. Wahrsch. Verw. Gebiete 47 119-137. · Zbl 0413.62024 · doi:10.1007/BF00535278
[9] 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. Deheuvels, P. (1977a). Estimation non paramétrique de la densité par histogrammes generalisés. Rev. Statist. Appl. 25 5-42. Deheuvels, P. (1977b). Estimation nonparamétrique de la densité par histogrammes generalisés. Publ. Inst. Statist. Univ. Paris 22 1-23. · Zbl 0937.62518 · doi:10.1016/0167-9473(92)00066-Z
[10] Deheuvels, P. and Hominal, P. (1980). Estimation automatique de la densité. Rev. Statist. Appl. 28 25-55. · Zbl 0503.62032
[11] Devroy e, L. (1983). The equivalence of weak, strong and complete convergence in L1 for kernel density estimates. Ann. Statist. 11 896-904. · Zbl 0521.62033 · doi:10.1214/aos/1176346255
[12] Devroy e, L. (1987). A Course in Density Estimation. Birkhäuser, Boston. · Zbl 0617.62043
[13] Devroy e, L. (1988). Asy mptotic performance bounds for the kernel estimate. Ann. Statist. 16 1162-1179. Devroy e, L. (1989a). A universal lower bound for the kernel estimate. Statist. Probab. Lett. 8 419-423. Devroy e, L. (1989b). The double kernel method in density estimation. Ann. Inst. H. Poincaré 25 533-580.
[14] Devroy e, L. (1991). Exponential inequalities in nonparametric estimation. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 31-44. Kluwer, Dordrecht. · Zbl 0739.62025
[15] Devroy e, L. (1992). A note on the usefulness of superkernels in density estimation. Ann. Statist. 20 2037-2056. · Zbl 0765.62038 · doi:10.1214/aos/1176348901
[16] Devroy e, L. (1994). On good deterministic smoothing sequences for kernel density estimates. Ann. Statist. 22 886-889. · Zbl 0805.62039 · doi:10.1214/aos/1176325500
[17] Devroy e, L. and Gy örfi, L. (1985). Nonparametric Density Estimation: The L1 View. Wiley, New York. · Zbl 0546.62015
[18] Devroy e, L. and Penrod, C. S. (1984). Distribution-free lower bounds in density estimation. Ann. Statist. 12 1250-1262. · Zbl 0551.62024 · doi:10.1214/aos/1176346790
[19] Donoho, D. L., Kerky acharian, G. and Picard, D. (1995). Wavelet shrinkage: asy mptotia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301-369. JSTOR:
[20] Donoho, D. L. and Liu, R. C. (1991). Geometrizing rates of convergence. III. Ann. Statist. 19 668-701. · Zbl 0754.62029 · doi:10.1214/aos/1176348115
[21] Hall, P. (1989). On convergence rates in nonparametric problems. Internat. Statist. Rev. 57 45-58. · Zbl 0707.62091 · doi:10.2307/1403583
[22] Hall, P. and Wand, M. P. (1988). Minimizing L1 distance in nonparametric density estimation. J. Multivariate Anal. 26 59-88. · Zbl 0673.62030 · doi:10.1016/0047-259X(88)90073-5
[23] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30. JSTOR: · Zbl 0127.10602 · doi:10.2307/2282952
[24] 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. · Zbl 0516.62043 · doi:10.1137/1127062
[25] Low, M. (1992). Non-existence of an adaptive estimator for the value of an unknown probability density. Ann. Statist. 20 598-602. · Zbl 0745.62035 · doi:10.1214/aos/1176348544
[26] Marron, J. S. (1988). Automatic smoothing parameter selection: a survey. Empirical Economics 13 187-208.
[27] Marron, J. S. (1989). Automatic smoothing parameter selection: a survey. In Semiparametric and Nonparametric Economics (A. Ullah, ed.) 65-86. Heidelberg.
[28] McDiarmid, C. (1989). On the method of bounded differences. In Survey s in Combinatorics 1989. London Math. Soc. Lecture Notes Ser. 141 148-188. Cambridge Univ. Press. · Zbl 0712.05012
[29] Park, B.-U. and Turlach, B. A. (1992). Practical performance of several data driven bandwidth selectors (with discussion). Comput. Statist. 7 251-270. · Zbl 0775.62100
[30] Parzen, E. (1962). On the estimation of a probability density function and the mode. Ann. Math. Statist. 33 1065-1076. · Zbl 0116.11302 · doi:10.1214/aoms/1177704472
[31] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 832-837. · Zbl 0073.14602 · doi:10.1214/aoms/1177728190
[32] Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scand. J. Statist. 9 65-78. · Zbl 0501.62028
[33] Stone, C. J. (1984). An asy mptotically optimal window selection rule for kernel density estimates. Ann. Statist. 12 1285-1297. · Zbl 0599.62052 · doi:10.1214/aos/1176346792
[34] Turlach, B. A. (1993). Bandwidth selection in kernel density estimation: a review. Technical report, Univ. Catholique de Louvain.
[35] Yatracos, Y. G. (1985). Rates of convergence of minimum distance estimators and Kolmogorov’s entropy. Ann. Statist. 13 768-774. · Zbl 0576.62057 · doi:10.1214/aos/1176349553
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.