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The bootstrap: Some large sample theory and connections with robustness

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  • Parr, William C.

Abstract

The bootstrap, discussed by Efron (1979, 1981), is a powerful tool for the nonparametric estimation of sampling distributions and asymptotic standard errors. We demonstrate consistency of the bootstrap distribution estimates for a general class of robust differentiable statistical functionals. Our conditions for consistency of the bootstrap are variants of previously considered criteria for robustness of the associated statistics. A general example shows that, for almost any location statistic, consistency of the bootstrap variance estimator requires a tail condition on the distribution from which samples are taken. A modification of Efron's estimator of standard error is shown to circumvent this problem.

Suggested Citation

  • Parr, William C., 1985. "The bootstrap: Some large sample theory and connections with robustness," Statistics & Probability Letters, Elsevier, vol. 3(2), pages 97-100, April.
    • Handle: RePEc:eee:stapro:v:3:y:1985:i:2:p:97-100
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    Citations

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    Cited by:

    1. Duarte Gonc{c}alves & Bruno A. Furtado, 2024. "Statistical Mechanism Design: Robust Pricing, Estimation, and Inference," Papers 2405.17178, arXiv.org.
    2. Jiménez Gamero, M. D. & Muñoz García, J. & Muñoz Reyes, A., 1998. "Bootstrapping statistical functionals," Statistics & Probability Letters, Elsevier, vol. 39(3), pages 229-236, August.
    3. M. Rajarshi, 1990. "Bootstrap in Markov-sequences based on estimates of transition density," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(2), pages 253-268, June.
    4. Cuevas, Antonio, 1992. "On the estimation of the influence curve," UC3M Working papers. Economics 2844, Universidad Carlos III de Madrid. Departamento de Economía.
    5. Salibian-Barrera, Matias, 2006. "Bootstrapping MM-estimators for linear regression with fixed designs," Statistics & Probability Letters, Elsevier, vol. 76(12), pages 1287-1297, July.
    6. Antonio Cuevas & Juan Romo, 1997. "Differentiable Functionals and Smoothed Bootstrap," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(2), pages 355-370, June.
    7. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2006. "On the use of the bootstrap for estimating functions with functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 1063-1074, November.
    8. Wei Luo & Bing Li, 2016. "Combining eigenvalues and variation of eigenvectors for order determination," Biometrika, Biometrika Trust, vol. 103(4), pages 875-887.

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