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Tenreiro, Carlos
Asymptotic behaviour of multistage plug-in bandwidth selections for kernel distribution function estimators. (English) Zbl 1087.62052
J. Nonparametric Stat. 18, No. 1, 101-116 (2006).
Summary: Given \(X_{1}, \dots, X_{n}\) independent real random variables with common but unknown absolutely continuous distribution function \(F\), we study the asymptotic behaviour of two classes of multistage plug-in bandwidth selectors for the kernel distribution function estimator \(\overline F_{n}\), on the basis of two asymptotic approximations of the optimal bandwidth \(h_{\text{MISE}}\) that minimize the mean integrated squared error \(E \int \{\overline F_{n}(x)-F(x)\}^{2}\,dx\). The second asymptotic approximation we consider is, to our knowledge, new in the literature. Although a better rate of convergence for \(h_{\text{MISE}}\) could be obtained by a multistage plug-in procedure based on this new asymptotic approximation, we prove that, from an asymptotic point of view, there is not a substantial difference between the two classes of associated kernel distribution function estimators in the sense of the integrated squared error.
For finite sample sizes, a simulation study indicates that the plug-in methods based on the new asymptotic approximation of the optimal bandwidth are superior to the corresponding one based on the asymptotic approximation usually considered in the literature. Some comparisons with the cross-validation procedure proposed by A. Bowman, P. Hall and T. Prvan [Bandwidth selection for the smoothing of distribution functions. Biometrika 85, No. 4, 799–808 (1998; Zbl 0921.62042)] are also presented.

Cited in 1 Document
MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
Keywords:
kernel distribution function estimation; multistage plug-in bandwidth selection; asymptotic normality; mean integrated squared error; integrated squared error
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\textit{C. Tenreiro}, J. Nonparametric Stat. 18, No. 1, 101--116 (2006; Zbl 1087.62052)
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References:
[1] DOI: 10.1214/aoms/1177728190 · Zbl 0073.14602 · doi:10.1214/aoms/1177728190
[2] DOI: 10.1214/aoms/1177704472 · Zbl 0116.11302 · doi:10.1214/aoms/1177704472
[3] Tiago de Oliveira J., Revista da Faculdade de Ciências, Universidade de Lisboa Série A(2) 9 pp 111– (1963)
[4] Watson G. S., Sankhyā the Indian Journal of Statistics Series A 26 pp 101– (1964)
[5] DOI: 10.1137/1109069 · Zbl 0152.17605 · doi:10.1137/1109069
[6] Reiss R. D., Scandinavian Journal of Statistics 9 pp 65– (1981)
[7] DOI: 10.1080/03610928808829835 · Zbl 0696.62174 · doi:10.1080/03610928808829835
[8] DOI: 10.1016/0167-7152(92)90006-Q · Zbl 0686.62022 · doi:10.1016/0167-7152(92)90006-Q
[9] DOI: 10.1016/0167-9473(92)90061-J · Zbl 0937.62581 · doi:10.1016/0167-9473(92)90061-J
[10] DOI: 10.1016/0378-3758(93)90068-H · Zbl 0769.62030 · doi:10.1016/0378-3758(93)90068-H
[11] DOI: 10.1016/0378-3758(94)00102-2 · Zbl 0833.62035 · doi:10.1016/0378-3758(94)00102-2
[12] DOI: 10.1093/biomet/85.4.799 · Zbl 0921.62042 · doi:10.1093/biomet/85.4.799
[13] DOI: 10.1080/00949650008811990 · Zbl 0957.62031 · doi:10.1080/00949650008811990
[14] Martins, A. P. and Tenreiro, C. Multistage plug-in bandwidth selection for kernel distribution function estimators (in Portuguese with English abstract). Proceedings of the X Annual Meeting of Statistical Portuguese Society 2002. pp.415–426.
[15] DOI: 10.1080/10485250310001604631 · Zbl 1054.62061 · doi:10.1080/10485250310001604631
[16] DOI: 10.1016/S0167-7152(03)00176-7 · Zbl 1113.62317 · doi:10.1016/S0167-7152(03)00176-7
[17] Shirahata S., Annals of the Institute of Statistical Mathematics 44 pp 579– (1992)
[18] DOI: 10.1093/biomet/68.1.326 · doi:10.1093/biomet/68.1.326
[19] DOI: 10.1111/j.1467-9574.1983.tb00802.x · Zbl 0532.62024 · doi:10.1111/j.1467-9574.1983.tb00802.x
[20] DOI: 10.1093/biomet/78.2.263 · Zbl 0733.62045 · doi:10.1093/biomet/78.2.263
[21] DOI: 10.1007/BF01192160 · Zbl 0742.62041 · doi:10.1007/BF01192160
[22] DOI: 10.1016/0167-7152(87)90083-6 · Zbl 0628.62029 · doi:10.1016/0167-7152(87)90083-6
[23] DOI: 10.1016/0167-7152(91)90116-9 · Zbl 0724.62040 · doi:10.1016/0167-7152(91)90116-9
[24] Wand M. P., Kernel Smoothing (1995) · Zbl 0854.62043
[25] DOI: 10.2307/2684697 · doi:10.2307/2684697
[26] DOI: 10.1080/10485259208832524 · Zbl 1263.62059 · doi:10.1080/10485259208832524
[27] DOI: 10.1214/aos/1176325632 · Zbl 0821.62020 · doi:10.1214/aos/1176325632
[28] Gasser Th., Scandinavian Journal of Statistics 11 pp 197– (1985)
[29] Granovsky B. L., Statistical Decisions 13 pp 101– (1995)
[30] Silverman J., Density Estimation for Statistics and Data Analysis (1986) · Zbl 0617.62042 · doi:10.1007/978-1-4899-3324-9
[31] DOI: 10.1214/aos/1176348902 · Zbl 0765.62041 · doi:10.1214/aos/1176348902
[32] DOI: 10.1214/aos/1176348653 · Zbl 0746.62040 · doi:10.1214/aos/1176348653
[33] Bickel P. J., Sankhyā the Indian Journal of Statistics Series A 50 pp 381– (1988)
[34] DOI: 10.1007/978-1-4613-0125-7 · doi:10.1007/978-1-4613-0125-7
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