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Moment Consistency of the Exchangeably Weighted Bootstrap for Semiparametric M-estimation

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  • Guang Cheng

Abstract

type="main" xml:id="sjos12128-abs-0001"> The bootstrap variance estimate is widely used in semiparametric inferences. However, its theoretical validity is a well-known open problem. In this paper, we provide a first theoretical study on the bootstrap moment estimates in semiparametric models. Specifically, we establish the bootstrap moment consistency of the Euclidean parameter, which immediately implies the consistency of t-type bootstrap confidence set. It is worth pointing out that the only additional cost to achieve the bootstrap moment consistency in contrast with the distribution consistency is to simply strengthen the L 1 maximal inequality condition required in the latter to the L p maximal inequality condition for p≥1. The general L p multiplier inequality developed in this paper is also of independent interest. These general conclusions hold for the bootstrap methods with exchangeable bootstrap weights, for example, non-parametric bootstrap and Bayesian bootstrap. Our general theory is illustrated in the celebrated Cox regression model.

Suggested Citation

  • Guang Cheng, 2015. "Moment Consistency of the Exchangeably Weighted Bootstrap for Semiparametric M-estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(3), pages 665-684, September.
    • Handle: RePEc:bla:scjsta:v:42:y:2015:i:3:p:665-684
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    File URL: http://hdl.handle.net/10.1111/sjos.12128
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    References listed on IDEAS

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    1. Chen, Xiaohong & Pouzo, Demian, 2009. "Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals," Journal of Econometrics, Elsevier, vol. 152(1), pages 46-60, September.
    2. Murphy, S. A. & van der Vaart, A. W., 2001. "Semiparametric Mixtures in Case-Control Studies," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 1-32, October.
    3. Yoichi Nishiyama, 2010. "Moment convergence of M‐estimators," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(4), pages 505-507, November.
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    2. Jean-Jacques Forneron, 2022. "Estimation and Inference by Stochastic Optimization," Papers 2205.03254, arXiv.org.
    3. Forneron, Jean-Jacques, 2024. "Estimation and inference by stochastic optimization," Journal of Econometrics, Elsevier, vol. 238(2).
    4. Vanhems, Anne & Van Keilegom, Ingrid, 2019. "Estimation Of A Semiparametric Transformation Model In The Presence Of Endogeneity," Econometric Theory, Cambridge University Press, vol. 35(1), pages 73-110, February.
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    7. Inass Soukarieh & Salim Bouzebda, 2022. "Exchangeably Weighted Bootstraps of General Markov U -Process," Mathematics, MDPI, vol. 10(20), pages 1-42, October.

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