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Article

Central limit theorems for functional Z-estimators with functional nuisance parameters

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Pages 2535-2577 | Received 22 Jul 2022, Accepted 13 Oct 2022, Published online: 04 Nov 2022

References

  • Akritas, M. G. 1986. Bootstrapping the Kaplan-Meier estimator. Journal of the American Statistical Association 81 (396):1032–8.
  • Aldous, D. J. 1985. Exchangeability and related topics. In École d’été de probabilités de Saint-Flour, XIII—1983, volume 1117 of Lecture Notes in Math, 1–198. Berlin: Springer.
  • Alvarez-Andrade, S, and S. Bouzebda. 2015. On the local time of the weighted bootstrap and compound empirical processes. Stochastic Analysis and Applications 33 (4):609–29. doi:10.1080/07362994.2015.1024854.
  • Banerjee, M., D. Mukherjee, and S. Mishra. 2009. Semiparametric binary regression models under shape constraints with an application to Indian schooling data. Journal of Econometrics 149 (2):101–17. doi:10.1016/j.jeconom.2008.11.002.
  • Barbe, P, and P. Bertail. 1995. The weighted bootstrap, volume 98 of Lecture Notes in Statistics. New York: Springer-Verlag.
  • Beran, R. 2003. The impact of the bootstrap on statistical algorithms and theory. Statistical Science 18 (2):175–84. Silver anniversary of the bootstrap. doi:10.1214/ss/1063994972.
  • Bickel, P. J, and D. A. Freedman. 1981. Some asymptotic theory for the bootstrap. The Annals of Statistics 9 (6):1196–217. doi:10.1214/aos/1176345637.
  • Bickel, P. J., F. Götze, and W. R. van Zwet. 1997. Resampling fewer than n observations: Gains, losses, and remedies for losses. Statistica Sinica 7 (1):1–31. Empirical Bayes, sequential analysis and related topics in statistics and probability (New Brunswick, NJ, 1995).
  • Bickel, P. J., C. A. J. Klaassen, Y. Ritov, and J. A. Wellner. 1998. Efficient and adaptive estimation for semiparametric models. New York: Springer-Verlag. Reprint of the 1993 original.
  • Billingsley, P. 1968. Convergence of probability measures. New York-London-Sydney: John Wiley & Sons, Inc.
  • Bouzebda, S. 2012. On the strong approximation of bootstrapped empirical copula processes with applications. Mathematical Methods of Statistics 21 (3):153–88. doi:10.3103/S1066530712030015.
  • Bouzebda, S. 2014. Asymptotic properties of pseudo maximum likelihood estimators and test in semi-parametric copula models with multiple change points. Mathematical Methods of Statistics 23 (1):38–65. doi:10.3103/S1066530714010037.
  • Bouzebda, S, and M. Cherfi. 2012. General bootstrap for dual ϕ-divergence estimates. Journal of Probability and Statistics. Art. ID 834107:33.
  • Bouzebda, S, and A. A. Ferfache. 2021. Asymptotic properties of M-estimators based on estimating equations and censored data in semi-parametric models with multiple change points. Journal of Mathematical Analysis and Applications. 497 (2):124883. Paper No. 124883, 44. doi:10.1016/j.jmaa.2020.124883.
  • Bouzebda, S, and A. Keziou. 2010a. New estimates and tests of independence in semiparametric copula models. Kybernetika (Prague) 46 (1):178–201.
  • Bouzebda, S, and A. Keziou. 2010b. A new test procedure of independence in copula models via χ2-divergence. Communications in Statistics - Theory and Methods 39 (1–2):1–20.
  • Bouzebda, S, and A. Keziou. 2013. A semiparametric maximum likelihood ratio test for the change point in copula models. Statistical Methodology 14:39–61. doi:10.1016/j.stamet.2013.02.003.
  • Bouzebda, S, and N. Limnios. 2013. On general bootstrap of empirical estimator of a semi-Markov kernel with applications. Journal of Multivariate Analysis. 116:52–62. doi:10.1016/j.jmva.2012.11.008.
  • Bouzebda, S., C. Papamichail, and N. Limnios. 2018. On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications. Journal of Nonparametric Statistics 30 (1):49–86. doi:10.1080/10485252.2017.1404059.
  • Bouzebda, S., I. Elhattab, and A. A. Ferfache. 2022. General M-estimator processes and their m out of n bootstrap with functional nuisance parameters. In Methodology and computing in applied probability, 1–45. doi:10.1007/s11009-022-09965-y.
  • Cheng, G. 2009. Semiparametric additive isotonic regression. Journal of Statistical Planning and Inference 139 (6):1980–91. doi:10.1016/j.jspi.2008.09.009.
  • Cheng, G. 2015. Moment consistency of the exchangeably weighted bootstrap for semiparametric M-estimation. Scandinavian Journal of Statistics 42 (3):665–84. doi:10.1111/sjos.12128.
  • Cheng, G, and J. Z. Huang. 2010. Bootstrap consistency for general semiparametric M-estimation. The Annals of Statistics 38 (5):2884–915. doi:10.1214/10-AOS809.
  • Cheng, R. 2017. Non-standard parametric statistical inference. Oxford: Oxford University Press.
  • Chernick, M. R. 1999. Bootstrap methods. In Wiley Series in Probability and Statistics: Applied Probability and Statistics. New York: John Wiley & Sons, Inc. A practitioner’s guide, A Wiley-Interscience Publication.
  • Chernick, M. R. 2008. Bootstrap methods: A guide for practitioners and researchers. In Wiley series in probability and statistics. 2nd ed. Hoboken, NJ: Wiley-Interscience [John Wiley & Sons].
  • Davison, A. C, and D. V. Hinkley. 1997. Bootstrap methods and their application, volume 1 of Cambridge Series in Statistical and probabilistic mathematics. Cambridge: Cambridge University Press. With 1 IBM-PC floppy disk (3.5 inch; HD).
  • Dunford, N, and J. T. Schwartz. 1958. Linear operators part I: General theory, volume 243. New York: Interscience Publishers.
  • Dunford, N, and J. T. Schwartz. 1988. Linear operators. Part I. Wiley Classics Library. New York: John Wiley & Sons, Inc. General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication.
  • Efron, B. 1979. Bootstrap methods: Another look at the jackknife. The Annals of Statistics 7 (1):1–26. doi:10.1214/aos/1176344552.
  • Gill, R. D. 1989. Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. Scandinavian Journal of Statistics 16 (2):97–128. With a discussion by J. A. Wellner and J. Praestgaard and a reply by the author.
  • Giné, E, and J. Zinn. 1990. Bootstrapping general empirical measures. The Annals of Probability 18 (2):851–69. doi:10.1214/aop/1176990862.
  • Good, P. 2005. Permutation, parametric and bootstrap tests of hypotheses. 3rd ed. New York: Springer Series in Statistics. Springer-Verlag.
  • Gu, M, and C.-H. Zhang. 1993. Asymptotic properties of self-consistent estimators based on doubly censored data. The Annals of Statistics 21 (2):611–24. doi:10.1214/aos/1176349140.
  • Hall, P. 1992. The bootstrap and Edgeworth expansion. New York: Springer Series in Statistics. Springer-Verlag.
  • Huang, J. 1999. Efficient estimation of the partly linear additive Cox model. The Annals of Statistics 27 (5):1536–63. doi:10.1214/aos/1017939141.
  • Huber, P. J. 1964. Robust estimation of a location parameter. The Annals of Mathematical Statistics 35 (1):73–101. doi:10.1214/aoms/1177703732.
  • Huber, P. J. 1967. The behavior of maximum likelihood estimates under nonstandard conditions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. I: Statistics, 221–33. Berkeley, CA: Univ. California Press.
  • James, L. F. 1997. A study of a class of weighted bootstraps for censored data. The Annals of Statistics 25 (4):1595–621. doi:10.1214/aos/1031594733.
  • Janssen, A. 2005. Resampling student’s t-type statistics. Annals of the Institute of Statistical Mathematics 57 (3):507–29. doi:10.1007/BF02509237.
  • Janssen, A, and T. Pauls. 2003. How do bootstrap and permutation tests work? The Annals of Statistics 31 (3):768–806. doi:10.1214/aos/1056562462.
  • Kallenberg, O. 2002. Foundations of modern probability. In Probability and its applications. 2nd ed. New York: Springer-Verlag.
  • Kaplan, E. L, and P. Meier. 1958. Nonparametric estimation from incomplete observations. Journal of the American Statistical Association 53 (282):457–81. doi:10.1080/01621459.1958.10501452.
  • Kosorok, M. R. 2008. Introduction to empirical processes and semiparametric inference. In Springer series in statistics. New York: Springer.
  • Kosorok, M. R. 2009. What’s so special about semiparametric methods? Sankhyā 71 (2, Ser. A):331–53.
  • Lehmann, E. L, and G. Casella. 1998. Theory of point estimation. In Springer texts in statistics. 2nd ed. New York: Springer-Verlag.
  • Lehmann, E. L, and J. P. Romano. 2005. Testing statistical hypotheses. In Springer texts in statistics. 3rd ed. New York: Springer.
  • Lindsey, J. K. 1996. Parametric statistical inference. New York: Oxford Science Publications. The Clarendon Press, Oxford University Press.
  • Lo, A. Y. 1993. A Bayesian method for weighted sampling. The Annals of Statistics 21 (4):2138–48. doi:10.1214/aos/1176349414.
  • Ma, S, and M. R. Kosorok. 2005. Robust semiparametric M-estimation and the weighted bootstrap. Journal of Multivariate Analysis. 96 (1):190–217. doi:10.1016/j.jmva.2004.09.008.
  • Manly, B. F. J. 2007. Randomization, bootstrap and Monte Carlo methods in biology. In Chapman & Hall/CRC texts in statistical science series. 3rd ed. Boca Raton, FL: Chapman & Hall/CRC.
  • Mason, D. M, and M. A. Newton. 1992. A rank statistics approach to the consistency of a general bootstrap. The Annals of Statistics 20 (3):1611–24. doi:10.1214/aos/1176348787.
  • Murphy, S. A. 1995. Asymptotic theory for the frailty model. The Annals of Statistics 23 (1):182–98. doi:10.1214/aos/1176324462.
  • Pakes, A, and D. Pollard. 1989. Simulation and the asymptotics of optimization estimators. Econometrica 57 (5):1027–57. doi:10.2307/1913622.
  • Parner, E. 1998. Asymptotic theory for the correlated gamma-frailty model. The Annals of Statistics 26 (1):183–214. doi:10.1214/aos/1030563982.
  • Pauly, M. 2012. Consistency of the subsample bootstrap empirical process. Statistics 46 (5):621–6. doi:10.1080/02331888.2010.543469.
  • Pfanzagl, J. 1994. Parametric statistical theory. Berlin: De Gruyter Textbook. Walter de Gruyter & Co. With the assistance of R. Hamböker.
  • Pollard, D. 1985. New ways to prove central limit theorems. Econometric Theory 1 (3):295–313. doi:10.1017/S0266466600011233.
  • Pollard, D. 1989. Asymptotics via empirical processes. Statistical Science 4 (4):341–66. With comments and a rejoinder by the author. doi:10.1214/ss/1177012399.
  • Praestgaard, J, and J. A. Wellner. 1993. Exchangeably weighted bootstraps of the general empirical process. The Annals of Probability 21 (4):2053–86. doi:10.1214/aop/1176989011.
  • Rubin, D. B. 1981. The Bayesian bootstrap. The Annals of Statistics 9 (1):130–4. doi:10.1214/aos/1176345338.
  • Shao, J, and D. S. Tu. 1995. The jackknife and bootstrap. In Springer series in statistics. New York: Springer-Verlag.
  • Shao, J, and C. Wu. 1987. Heteroscedasticity-robustness of jackknife variance estimators in linear models. The Annals of Statistics 15 (4):1563–79. doi:10.1214/aos/1176350610.
  • Tsai, W.-Y, and C.-H. Zhang. 1995. Asymptotic properties of nonparametric maximum likelihood estimator for interval-truncated data. Scandinavian Journal of Statistics 22 (3):361–70.
  • van der Laan, M. J. 1995. Efficient and inefficient estimation in semiparametric models, volume 114 of CWI Tract. Amsterdam: Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica.
  • van der Vaart, A. 1994. Maximum likelihood estimation with partially censored data. The Annals of Statistics 22 (4):1896–916. doi:10.1214/aos/1176325763.
  • Vaart, A. W. 1995. Efficiency of infinite-dimensional M-estimators. Statistica Neerlandica 49 (1):9–30. doi:10.1111/j.1467-9574.1995.tb01452.x.
  • van der Vaart, A. W. 1998. Asymptotic statistics, volume 3 of Cambridge series in statistical and probabilistic mathematics. Cambridge: Cambridge University Press.
  • van der Vaart, A. W, and J. A. Wellner. 1996. Weak convergence and empirical processes. In Springer series in statistics. New York: Springer-Verlag. With applications to statistics.
  • van Zwet, W. R. 1979. The Edgeworth expansion for linear combinations of uniform order statistics. In Proceedings of the Second Prague Symposium on Asymptotic Statistics (Hradec Králové, 1978), 93–101. Amsterdam-New York: North-Holland.
  • Vardi, Y, and C.-H. Zhang. 1992. Large sample study of empirical distributions in a random-multiplicative censoring model. The Annals of Statistics 20 (2):1022–39. doi:10.1214/aos/1176348668.
  • Wellner, J. A, and Y. Zhan. 1996. Bootstrapping Z-estimators. University of Washington Department of Statistics Technical Report 308:5.
  • Wellner, J. A., C. A. J. Klaassen, and Y. Ritov. 2006. Semiparametric models: A review of progress since BKRW (1993). In Frontiers in statistics, 25–44. London: Imp. Coll. Press.
  • Weng, C.-S. 1989. On a second-order asymptotic property of the Bayesian bootstrap mean. The Annals of Statistics 17 (2):705–10. doi:10.1214/aos/1176347136.
  • Zeng, D, and D. Y. Lin. 2007. Maximum likelihood estimation in semiparametric regression models with censored data. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69 (4):507–64. With discussion and a reply by the authors. doi:10.1111/j.1369-7412.2007.00606.x.
  • Zhan, Y. 1996. Bootstrapping functional M-estimators., ProQuest LLC, Ann Arbor, MI. Thesis (PhD). University of Washington.
  • Zhan, Y. 2002. Central limit theorems for functional Z-estimators. Statistics Sinica 12 (2):609–34.
  • Zhang, C, and T. Yu. 2008. Semiparametric detection of significant activation for brain fMRI. The Annals of Statistics 36 (4):1693–725. doi:10.1214/07-AOS519.
  • Zheng, Z. G, and D. S. Tu. 1988. Random weighting methods in regression models. Science in China (Scientia Sinica) Series A 31 (12):1442–59.

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