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Bootstraps of sums of independent but not identically distributed stochastic processes

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  • Kosorok, Michael R.

Abstract

A central limit theorem is developed for sums of independent but not identically distributed stochastic processes multiplied by independent real random variables with mean zero. Weak convergence of the Hoffmann-Jørgensen-Dudley type, as described in van der Vaart and Wellner (Weak Convergence and Empirical Processes, Springer, New York, 1996), is utilized. These results allow Monte Carlo estimation of limiting probability measures obtained from application of Pollard's (Empirical Processes: Theory and Applications, IMS, Hayward, CA, 1990) functional central limit theorem for empirical processes. An application of this theory to the two-parameter Cox score process with staggered entry data is given for illustration. For this process, the proposed multiplier bootstrap appears to be the first successful method for estimating the associated limiting distribution. The results of this paper compliment previous bootstrap and multiplier central limit theorems for independent and identically distributed empirical processes.

Suggested Citation

  • Kosorok, Michael R., 2003. "Bootstraps of sums of independent but not identically distributed stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 299-318, February.
    • Handle: RePEc:eee:jmvana:v:84:y:2003:i:2:p:299-318
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    References listed on IDEAS

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    1. Ziegler, Klaus, 1997. "Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions," Journal of Multivariate Analysis, Elsevier, vol. 62(2), pages 233-272, August.
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    Cited by:

    1. González-Rodríguez, Gil & Colubi, Ana, 2017. "On the consistency of bootstrap methods in separable Hilbert spaces," Econometrics and Statistics, Elsevier, vol. 1(C), pages 118-127.
    2. Beare, Brendan K. & Seo, Juwon, 2020. "Randomization Tests Of Copula Symmetry," Econometric Theory, Cambridge University Press, vol. 36(6), pages 1025-1063, December.
    3. Zhengyu Zhang & Zequn Jin & Lihua Lin, 2024. "Identification and inference of outcome conditioned partial effects of general interventions," Papers 2407.16950, arXiv.org.
    4. Chiang, Harold D. & Hsu, Yu-Chin & Sasaki, Yuya, 2019. "Robust uniform inference for quantile treatment effects in regression discontinuity designs," Journal of Econometrics, Elsevier, vol. 211(2), pages 589-618.
    5. Hoffmann, Michael & Vetter, Mathias & Dette, Holger, 2018. "Nonparametric inference of gradual changes in the jump behaviour of time-continuous processes," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3679-3723.
    6. Telschow, Fabian J.E. & Davenport, Samuel & Schwartzman, Armin, 2022. "Functional delta residuals and applications to simultaneous confidence bands of moment based statistics," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    7. Andreas Hagemann, 2017. "Cluster-Robust Bootstrap Inference in Quantile Regression Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 446-456, January.
    8. D’Haultfœuille, Xavier & Hoderlein, Stefan & Sasaki, Yuya, 2024. "Testing and relaxing the exclusion restriction in the control function approach," Journal of Econometrics, Elsevier, vol. 240(2).
    9. Chang, Chung & Todd Ogden, R., 2009. "Bootstrapping sums of independent but not identically distributed continuous processes with applications to functional data," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1291-1303, July.
    10. Hongtu Zhu & Joseph G. Ibrahim & Xiaoyan Shi, 2009. "Diagnostic Measures for Generalized Linear Models with Missing Covariates," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 686-712, December.

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