Weak convergence for stationary bootstrap empirical processes of associated sequences
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this work the stationary bootstrap of Politis and Romano \cite{PR1994a} is applied to the empirical distribution function of stationary and associated random variables. A weak convergence theorem for the stationary bootstrap empirical processes of associated sequences is established with its limiting to a Gaussian process almost surely, conditionally on the stationary observations. The weak convergence result is proved by means of a random central limit theorem on geometrically distributed random block size of the stationary bootstrap procedure. As its statistical applications, stationary bootstrap quantiles and stationary bootstrap mean residual life process are discussed. Our results extend the existing ones of Peligrad \cite{P1998} who dealt with the weak convergence of non-random blockwise empirical processes of associated sequences as well as of Shao and Yu \cite{SY1996} who obtained the weak convergence of the mean residual life process in reliability theory as an application of the association.
Abstract
In this work the stationary bootstrap of Politis and Romano \cite{PR1994a} is applied to the empirical distribution function of stationary and associated random variables. A weak convergence theorem for the stationary bootstrap empirical processes of associated sequences is established with its limiting to a Gaussian process almost surely, conditionally on the stationary observations. The weak convergence result is proved by means of a random central limit theorem on geometrically distributed random block size of the stationary bootstrap procedure. As its statistical applications, stationary bootstrap quantiles and stationary bootstrap mean residual life process are discussed. Our results extend the existing ones of Peligrad \cite{P1998} who dealt with the weak convergence of non-random blockwise empirical processes of associated sequences as well as of Shao and Yu \cite{SY1996} who obtained the weak convergence of the mean residual life process in reliability theory as an application of the association.
참고문헌 (37)
- Majid Asadi, "Properties and estimation of the mean past lifetime", Statistics, 46(3), pp.405~417 (2012), Informa UK Limited[DOI]
- P. Billingsley, "Convergence of Probability Measures" (1968), John Wiley & Sons, Inc
- P. B¨uhlmann, "Blockwise bootstrapped empirical process for stationary sequences", Ann. Statist, 22, pp.995~1012 (1994)[DOI]
- Peter Bühlmann, "The blockwise bootstrap for general empirical processes of stationary sequences", Stochastic Processes and their Applications, 58(2), pp.247~265 (1995), Elsevier BV[DOI]
- A. Bulinski, "Limit theorems for associated random fields and related systems" (2007), World Scientific Publishing Co. Pte. Ltd
- P. Doukhan, "Stochastic models for time series" (2018), Springer
- Giuseppe Cavaliere, "Dependent Wild Bootstrap for the Empirical Process", Journal of Time Series Analysis, 36(3), pp.290~314 (2015), Wiley[DOI]
- Paul Doukhan, "A new weak dependence condition and applications to moment inequalities", Stochastic Processes and their Applications, 84(2), pp.313~342 (1999), Elsevier BV[DOI]
- Paul Doukhan, "Functional central limit theorem for the empirical process of short memory linear processes", Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 326(1), pp.87~92 (1998), Elsevier BV[DOI]
- Farid El Ktaibi, "Bootstrapping the empirical distribution of a linear process", Statistics & Probability Letters, 93, pp.134~142 (2014), Elsevier BV[DOI]
- Farid El Ktaibi, "Bootstrapping the empirical distribution of a stationary process with change-point", Electronic Journal of Statistics, 13(2), pp.3572~3612 (2019), Institute of Mathematical Statistics[DOI]
- J. K. Ghosh, "A new proof of the Bahadur representation of quantiles and an application", Ann. Math. Statist, 42, pp.1957~1961 (1971)[DOI]
- Eunju Hwang, "Stationary bootstrapping for non-parametric estimator of nonlinear autoregressive model", Journal of Time Series Analysis, 32(3), pp.292~303 (2011), Wiley[DOI]
- E. Hwang, "Strong consistency of the stationary bootstrap under ψ-weak dependence", Stat. Probab. Lett, 82, pp.488~495 (2012)
- 황은주, "Random central limit theorems for linear processes with weakly dependent innovations", Journal of the Korean Statistical Society, 41(3), pp.313~322 (2012), 한국통계학회
- E. Hwang, "Stationary bootstrapping realized volatility under market microstructure noise", Electron. J. Stat, 7, pp.2032~2053 (2013)
- E. Hwang, "Stationary bootstrapping realized volatility", Stat. Probab. Lett, 83, pp.2045~2051 (2013)
- Eunju Hwang, "Stationary bootstrapping for semiparametric panel unit root tests", Computational Statistics & Data Analysis, 83, pp.14~25 (2015), Elsevier BV[DOI]
- H. R. K¨unsch, "The jackknife and the bootstrap for general stationary observations", Ann. Statist, 17(3), pp.1217~1241 (1989)[DOI]
- Sangyeol Lee, "Random central limit theorem for the linear process generated by a strong mixing process", Statistics & Probability Letters, 35(2), pp.189~196 (1997), Elsevier BV[DOI]
- S Louhichi, "Weak convergence for empirical processes of associated sequences", Annales de l'Institut Henri Poincare (B) Probability and Statistics, 36(5), pp.547~567 (2000), Elsevier BV[DOI]
- U. V. Naik-Nimbalkar, "Validity of blockwise bootstrap for empirical processes with stationary observations", Ann. Statist, 22(2), pp.980~994 (1994)[DOI]
- M. Shafaei Noughabi, "Bivariate quantile residual life: a characterization theorem and statistical properties", Statistical Papers, 60(6), pp.2001~2012 (2019), Springer Science and Business Media LLC[DOI]
- Afshin Parvardeh, "A note on the asymptotic distribution of the estimation of the mean past lifetime", Statistical Papers, 56(1), pp.205~215 (2015), Springer Science and Business Media LLC[DOI]
- M. Peligrad, "On the blockwise bootstrap for empirical processes for stationary sequences", Ann. Probab, 26(2), pp.877~901 (1998)[DOI]
- D. N. Politis, "The stationary bootstrap", J. Amer. Statist. Assoc, 89(428), pp.1303~1313 (1994)
- D. N. Politis, "Limit theorems for weakly dependent Hilbert space valued random variables with applications to the stationary bootstrap", Stat. Sin, 4, pp.461~476 (1994)
- D. Pollard, "Convergence of Stochastic Processes, Springer Series in Statistics" (1984), SpringerVerlag
- Dragan Radulović, "The bootstrap for empirical processes based on stationary observations", Stochastic Processes and their Applications, 65(2), pp.259~279 (1996), Elsevier BV[DOI]
- D. Radulovi´c, "High Dimensional Probability", pp.315~330 (1998), Birkh¨auser
- George G. Roussas, "Kernel estimates under association: strong uniform consistency", Statistics & Probability Letters, 12(5), pp.393~403 (1991), Elsevier BV[DOI]
- Xiaofeng Shao, "The Dependent Wild Bootstrap", Journal of the American Statistical Association, 105(489), pp.218~235 (2010), Informa UK Limited[DOI]
- Qi-Man Shao, "Bootstrapping the sample means for stationary mixing sequences", Stochastic Processes and their Applications, 48(1), pp.175~190 (1993), Elsevier BV[DOI]
- Q. M. Shao, "Weak convergence for weighted empirical processes of dependent sequences", Ann. Probab, 24(4), pp.2098~2127 (1996)[DOI]
- Barbara Wieczorek, "Blockwise bootstrap of the estimated empirical process based on ψ-weakly dependent observations", Statistical Inference for Stochastic Processes, 19(1), pp.111~129 (2016), Springer Science and Business Media LLC[DOI]
- Hao Yu, "A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences", Probability Theory and Related Fields, 95(3), pp.357~370 (1993), Springer Science and Business Media LLC[DOI]