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Local bootstrap

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Abstract

The local bootstrap method draws bootstrap samples from a neighborhood of each data point. Unlike the common bootstrap, it is especially useful for heteroscedastic data. This paper considers local bootstrap of y for a data set {(x i , y i )} with the neighborhood defined by the window in a kernel regression model. It shows that for each x, the bootstrap distribution and moments converge almost surely to the true distribution and moments of y respectively, and this convergence is also uniform in x within the data set.

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References

  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.

    Google Scholar 

  • Cheng, K. F. and Lin, P. E. (1981) Nonparametric estimates of a regression function, Z. Wahrsch. Verw. Gebiete, 57, 223–233.

    Google Scholar 

  • Efron, B. (1970). Bootstrap methods: another look at the jackknife, Ann. Statist., 7, 1–26.

    Google Scholar 

  • Gasser, T. and Müller, H.-G. (1979). Kernel estimation of regression functions, Smoothing Techniques for Curve Estimation, Lecture Notes in Math., 757, 23–68, Springer, Berlin.

    Google Scholar 

  • Härdle, W. and Brown, A. (1988). Bootstrapping in nonparametric regression: local adaptive smoothing and confidence bands, J. Amer. Statist. Assoc., 83, 102–110.

    Google Scholar 

  • Hinkley, D. V. (1986). Comment on “Jackknife, bootstrap and other resampling methods in regression analysis”, Ann. Statist., 14, 1312–1316.

    Google Scholar 

  • Liu, R. Y. (1988). Bootstrap procedures under some non-i.i.d. models, Ann. Statist., 16, 1696–1708.

    Google Scholar 

  • Müller, H.-G. and Stadtmüller, U. (1987). Variable bandwidth kernel estimators of regression curves, Ann. Statist., 15, 182–201.

    Google Scholar 

  • Nadaraya, E. A. (1964). On estimating regression, Theory Probab. Appl., 9, 141–142.

    Google Scholar 

  • Priestley, M. B. and Chao, M. T. (1972). Nonparametric function fitting, J. Roy. Statist. Soc. Ser. B, 34, 385–392.

    Google Scholar 

  • Watson, G. S. (1964). Smooth regression analysis, Sankhya Sev. A, 26, 350–379.

    Google Scholar 

  • Wu, C. F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis (with discussion), Ann. Statist., 14, 1261–1350.

    Google Scholar 

Download references

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Author notes

  1. Sheng G. Shi

    Present address: Research, PRUPAC, 23 Main Street, P.O. Box 419, 07733, Holmdel, NJ, U.S.A.

Authors and Affiliations

  1. Department of Mathematical Science, Oakland University, 48309, Rochester, MI, U.S.A.

    Sheng G. Shi

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  1. Sheng G. Shi
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Shi, S.G. Local bootstrap. Ann Inst Stat Math 43, 667–676 (1991). https://doi.org/10.1007/BF00121646

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  • DOI: https://doi.org/10.1007/BF00121646

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