×

zbMATH — the first resource for mathematics

The choice of smoothing parameter in nonparametric regression through wild bootstrap. (English) Zbl 1429.62139
Summary: A bootstrap method to estimate the mean squared error and the smoothing parameter for the multidimensional regression local linear estimator is proposed. This method is based on resampling of the estimated residuals. It uses a bootstrap estimator of the mean squared error to select an asymptotically optimal bandwidth parameter. This is achieved by showing that the mean squared error and its bootstrap estimator are very closed. Thus, the smoothing parameter minimizing the mean squared error is asymptotically close to the smoothing parameter minimizing the bootstrap estimator of the mean squared error. The results are extended to the case in which the response variable contains missing observations.

MSC:
62G08 Nonparametric regression and quantile regression
62G07 Density estimation
62G09 Nonparametric statistical resampling methods
62G20 Asymptotic properties of nonparametric inference
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cao-Abad, R., Rate of convergence for the wild bootstrap in nonparametric regression, Ann. statist, 19, 4, 2226-2231, (1991) · Zbl 0745.62038
[2] Cao-Abad, R., Bootstrapping the Mean integrated squared error, J. multivariate anal, 45, 1, 137-160, (1993) · Zbl 0779.62038
[3] Chu, C.K.; Cheng, P.E., Nonparametric regression estimation with missing data, J. statist. plann. inference, 48, 1, 85-99, (1995) · Zbl 0897.62038
[4] Delaigle, A.; Gijbels, I., Practical bandwidth selection in deconvolution kernel density estimation, Comput. statist. data anal, 45, 2, 249-267, (2004) · Zbl 1429.62125
[5] Efron, B., Bootstrap methods: another look at the jackknife, Ann. statist, 7, 1, 1-26, (1979) · Zbl 0406.62024
[6] Efron, B., Missing data, imputation, and the bootstrap, J. amer. statist. assoc, 89, 426, 463-479, (1994) · Zbl 0806.62033
[7] Fan, J.; Gijbels, I., Data-driven bandwidth selection in local polynomial fittingvariable bandwidth and spatial adaptation, J. roy. statist. soc. ser. B, 57, 2, 371-394, (1995) · Zbl 0813.62033
[8] González Manteiga, W.; Pérez González, A., Nonparametric Mean estimation with missing data, Commun. statist, 33, 2, 277-303, (2004) · Zbl 1102.62033
[9] Hall, P., Using the bootstrap to estimate Mean squared error and select smoothing parameter in nonparametric problems, J. multivariate anal, 32, 2, 177-203, (1990) · Zbl 0722.62030
[10] Härdle, W.; Huet, S.; Jolivet, E., Better bootstrap confidence intervals for regression curve estimation, Statistics, 26, 4, 287-306, (1995) · Zbl 0836.62031
[11] Härdle, W.; Mammen, E., Comparing nonparametric versus parametric regression fits, Ann. statist, 21, 4, 1926-1947, (1993) · Zbl 0795.62036
[12] Härdle, W.; Marron, J.S., Bootstrap simultaneous error bars for nonparametric regression, Ann. statist, 19, 2, 778-796, (1991) · Zbl 0725.62037
[13] Little, R.J.A.; Rubin, D.B., Statistical analysis with missing data. wiley series in probability and mathematical statistics: applied probability and statistics, (2002), Wiley New York
[14] Mammen, E., 2000. Resampling methods for nonparametric regression. In: Schimek, M.G. (Ed.), Smoothing and Regression: Approaches, Computation and Application. Wiley, New York. · Zbl 0980.62031
[15] Nadaraya, E.A., On estimating regression, Theory probab. appl, 10, 186-190, (1964) · Zbl 0134.36302
[16] Neumann, M., Pointwise confidence intervals in nonparametric regression with heteroscedastic error structure, Statistics, 29, 1, 1-36, (1997) · Zbl 0869.62034
[17] Neumann, M.; Polzehl, J., Simultaneous bootstrap confidence bands in nonparametric regression, J. nonparametr. statist, 9, 4, 307-333, (1998) · Zbl 0913.62041
[18] Ruppert, D., Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation, J. amer. statist. assoc, 92, 439, 1049-1062, (1997) · Zbl 1067.62531
[19] Ruppert, D.; Wand, M.P., Multivariate locally weighted least squares regression, Ann. statist, 22, 3, 1346-1370, (1994) · Zbl 0821.62020
[20] Saavedra, A.; Cao-Abad, R., Smoothed bootstrap bandwidth selection in nonparametric density estimation for moving average processes, Estochastic anal. appl, 19, 4, 555-580, (2001) · Zbl 0980.62026
[21] Shao, J.; Tu, D., The jackknife and bootstrap. Springer series in statistics, (1995), Springer New York
[22] Stute, W.; González Manteiga, W.; Presedo Quindimil, M., Bootstrap approximations in model checks for regression, J. amer. statist. assoc, 93, 441, 141-149, (1998) · Zbl 0902.62027
[23] Vieu, P., Nonparametric regressionoptimal local bandwidth choice, J. roy. statist. soc. ser. B, 53, 2, 453-464, (1991) · Zbl 0800.62217
[24] Vieu, P., 1993. Bandwidth selection for kernel regression: a survey. In: Hardlë, W., Simar, L. (Eds.), Computer Intensive Methods in Statistics, Statistics and Computing, Vol. 1. Physica, Berlin, pp. 134-149.
[25] Watson, G.S., Smooth regression analysis, Sankhyā, ser. A, 26, 359-372, (1964) · Zbl 0137.13002
[26] Wu, C.-F.J., Jackknife, bootstrap and other resampling methods in regression analysis, Ann. statist, 14, 4, 1261-1350, (1986) · Zbl 0618.62072
[27] Yang, L.; Tschernig, R., Multivariate bandwidth selection for local linear regression, J. roy. statist. soc. ser. B stat. methodol, 61, 4, 793-815, (1999) · Zbl 0952.62039
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.