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Weighted bootstrapped kernel density estimators in two-sample problems

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  • Majid Mojirsheibani
  • William Pouliot

Abstract

A weighted bootstrap method is proposed to approximate the distribution of the $ L_p $ Lp ( $ 1\leq p<\infty $ 1≤p<∞) norms of two-sample statistics involving kernel density estimators. Using an approximation theorem of Horváth, Kozkoszka and Steineback [(2000) ‘Approximations for Weighted Bootstrap Processes with an Application’, Statistics and Probability Letters, 48, 59–70], that allows one to replace the weighted bootstrap empirical process by a sequence of Gaussian processes, we establish an unconditional bootstrap central limit theorem for such statistics. The proposed method is quite straightforward to implement in practice. Furthermore, through some simulation studies, it will be shown that, depending on the weights chosen, the proposed weighted bootstrap approximation can sometimes outperform both the classical large-sample theory as well as Efron's [(1979) ‘Bootstrap Methods: Another Look at the Jackknife’, Annals of Statistics, 7, 1–26] original bootstrap algorithm.

Suggested Citation

  • Majid Mojirsheibani & William Pouliot, 2017. "Weighted bootstrapped kernel density estimators in two-sample problems," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(1), pages 61-84, January.
    • Handle: RePEc:taf:gnstxx:v:29:y:2017:i:1:p:61-84
    DOI: 10.1080/10485252.2016.1253842
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    References listed on IDEAS

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    1. Hall, Peter, 1982. "Limit theorems for stochastic measures of the accuracy of density estimators," Stochastic Processes and their Applications, Elsevier, vol. 13(1), pages 11-25, July.
    2. Henze, Norbert & Nikitin, Yakov & Ebner, Bruno, 2009. "Integral distribution-free statistics of Lp-type and their asymptotic comparison," Computational Statistics & Data Analysis, Elsevier, vol. 53(9), pages 3426-3438, July.
    3. Burke, Murray D., 2000. "Multivariate tests-of-fit and uniform confidence bands using a weighted bootstrap," Statistics & Probability Letters, Elsevier, vol. 46(1), pages 13-20, January.
    4. Anderson, N. H. & Hall, P. & Titterington, D. M., 1994. "Two-Sample Test Statistics for Measuring Discrepancies Between Two Multivariate Probability Density Functions Using Kernel-Based Density Estimates," Journal of Multivariate Analysis, Elsevier, vol. 50(1), pages 41-54, July.
    5. Horváth, Lajos & Kokoszka, Piotr & Steinebach, Josef, 2000. "Approximations for weighted bootstrap processes with an application," Statistics & Probability Letters, Elsevier, vol. 48(1), pages 59-70, May.
    6. Hall, Peter, 1984. "Central limit theorem for integrated square error of multivariate nonparametric density estimators," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 1-16, February.
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    Cited by:

    1. Ali Al-Sharadqah & Majid Mojirsheibani & William Pouliot, 2020. "On the performance of weighted bootstrapped kernel deconvolution density estimators," Statistical Papers, Springer, vol. 61(4), pages 1773-1798, August.
    2. Ali Al-Sharadqah & Majid Mojirsheibani, 2020. "A simple approach to construct confidence bands for a regression function with incomplete data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(1), pages 81-99, March.

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