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Parametric bootstrapping with nuisance parameters

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  • Lee, Stephen M.S.
  • Young, G. Alastair

Abstract

Bootstrap methods are attractive empirical procedures for assessment of errors in problems of statistical estimation, and allow highly accurate inference in a vast range of parametric problems. Conventional parametric bootstrapping involves sampling from a fitted parametric model, obtained by substituting the maximum likelihood estimator for the unknown population parameter. Recently, attention has focussed on modified bootstrap methods which alter the sampling model used in the bootstrap calculation, in a systematic way that is dependent on the parameter of interest. Typically, inference is required for the interest parameter in the presence of a nuisance parameter, in which case the issue of how best to handle the nuisance parameter in the bootstrap inference arises. In this paper, we provide a general analysis of the error reduction properties of the parametric bootstrap. We show that conventional parametric bootstrapping succeeds in reducing error quite generally, when applied to an asymptotically normal pivot, and demonstrate further that systematic improvements are obtained by a particular form of modified scheme, in which the nuisance parameter is substituted by its constrained maximum likelihood estimator, for a given value of the parameter of interest.

Suggested Citation

  • Lee, Stephen M.S. & Young, G. Alastair, 2005. "Parametric bootstrapping with nuisance parameters," Statistics & Probability Letters, Elsevier, vol. 71(2), pages 143-153, February.
    • Handle: RePEc:eee:stapro:v:71:y:2005:i:2:p:143-153
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    References listed on IDEAS

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    1. Paul H. Garthwaite & Stephen T. Buckland, 1992. "Generating Monte Carlo Confidence Intervals by the Robbins–Monro Process," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 41(1), pages 159-171, March.
    2. J. Carpenter, 1999. "Test inversion bootstrap confidence intervals," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(1), pages 159-172.
    3. Stephen M. S. Lee, 2003. "Prepivoting by weighted bootstrap iteration," Biometrika, Biometrika Trust, vol. 90(2), pages 393-410, June.
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    Cited by:

    1. Robert L. Paige & A. Alexandre Trindade & P. Harshini Fernando, 2009. "Saddlepoint‐Based Bootstrap Inference for Quadratic Estimating Equations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 98-111, March.
    2. Godfrey, L.G., 2007. "Alternative approaches to implementing Lagrange multiplier tests for serial correlation in dynamic regression models," Computational Statistics & Data Analysis, Elsevier, vol. 51(7), pages 3282-3295, April.
    3. Lloyd, Chris J., 2012. "Computing highly accurate or exact P-values using importance sampling," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1784-1794.
    4. Guogen Shan & Changxing Ma, 2014. "Efficient tests for one sample correlated binary data with applications," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 23(2), pages 175-188, June.
    5. Lu, H.Y. Kevin & Young, G. Alastair, 2012. "Parametric bootstrap under model mis-specification," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2410-2420.
    6. Di Caterina, Claudia & Kosmidis, Ioannis, 2019. "Location-adjusted Wald statistics for scalar parameters," Computational Statistics & Data Analysis, Elsevier, vol. 138(C), pages 126-142.
    7. Lloyd, Chris J., 2010. "How close are alternative bootstrap P-values?," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1972-1976, December.
    8. Chris J. Lloyd, 2010. "Bootstrap and Second-Order Tests of Risk Difference," Biometrics, The International Biometric Society, vol. 66(3), pages 975-982, September.
    9. Lloyd, Chris J., 2013. "A numerical investigation of the accuracy of parametric bootstrap for discrete data," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 1-6.

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