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Saddlepoint‐Based Bootstrap Inference for Quadratic Estimating Equations

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  • ROBERT L. PAIGE
  • A. ALEXANDRE TRINDADE
  • P. HARSHINI FERNANDO

Abstract

. We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half‐century.

Suggested Citation

  • 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.
  • Handle: RePEc:bla:scjsta:v:36:y:2009:i:1:p:98-111
    DOI: 10.1111/j.1467-9469.2008.00614.x
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    References listed on IDEAS

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    1. Ronald W. Butler & Douglas A. Bronson, 2002. "Bootstrapping survival times in stochastic systems by using saddlepoint approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(1), pages 31-49, January.
    2. Malay Ghosh & Gauri Datta & Dalho Kim & Trevor Sweeting, 2006. "Likelihood-based Inference for the Ratios of Regression Coefficients in Linear Models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(3), pages 457-473, September.
    3. Lee, Stephen M.S. & Young, G. Alastair, 2005. "Parametric bootstrapping with nuisance parameters," Statistics & Probability Letters, Elsevier, vol. 71(2), pages 143-153, February.
    4. Simon Broda & Marc Paolella & Yianna Tchopourian, 2006. "Approximately Exact Inference in Dynamic Panel Models," Computing in Economics and Finance 2006 368, Society for Computational Economics.
    5. Andrews, Donald W K, 1993. "Exactly Median-Unbiased Estimation of First Order Autoregressive/Unit Root Models," Econometrica, Econometric Society, vol. 61(1), pages 139-165, January.
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    Cited by:

    1. Federico Martellosio & Grant Hillier, 2019. "Adjusted QMLE for the spatial autoregressive parameter," Papers 1909.08141, arXiv.org.
    2. Eugene Demidenko, 2017. "Exact and Approximate Statistical Inference for Nonlinear Regression and the Estimating Equation Approach," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(3), pages 636-665, September.
    3. Renren Zhao & Robert L. Paige, 2023. "Optimal equivalence testing in exponential families," Statistical Papers, Springer, vol. 64(5), pages 1507-1525, October.
    4. Robert Paige & A. Trindade & R. Wickramasinghe, 2014. "Extensions of saddlepoint-based bootstrap inference," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(5), pages 961-981, October.
    5. Martellosio, Federico & Hillier, Grant, 2020. "Adjusted QMLE for the spatial autoregressive parameter," Journal of Econometrics, Elsevier, vol. 219(2), pages 488-506.

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