Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics

Dehling, Herold; Sharipov, Olimjon Sh.; Wendler, Martin

doi:10.1016/j.jmva.2014.09.011

Cited by 26 publications

(19 citation statements)

References 32 publications

(35 reference statements)

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“…Our solution is the bootstrap, which has been successfully applied to many statistics in the case of real‐ or

R^{d}

‐valued data. For Hilbert spaces, only Politis & Romano () and recently Dehling, Sharipov, & Wendler () established the asymptotic validity of the bootstrap. The results of Politis & Romano () can only handle bounded random variables.…”

Section: Introductionmentioning

confidence: 99%

Sequential block bootstrap in a Hilbert space with application to change point analysis

2016

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A new test for structural changes in functional data is investigated. It is based on Hilbert space theory and critical values are deduced from bootstrap iterations. Thus a new functional central limit theorem for the block bootstrap in a Hilbert space is required. The test can also be used to detect changes in the marginal distribution of random vectors, which is supplemented by a simulation study. Our methods are applied to hydrological data from Germany. The Canadian Journal of Statistics 44: 300–322; 2016 © 2016 Statistical Society of Canada

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“…Our solution is the bootstrap, which has been successfully applied to many statistics in the case of real‐ or

R^{d}

Section: Introductionmentioning

confidence: 99%

Sequential block bootstrap in a Hilbert space with application to change point analysis

2016

Self Cite

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“…Thus, and by (2), for every 5 > 0, there exists m 5 ∈ N such that, for every m ≥ m 5 , this term is bounded by 5 . For the last term of (18), note that { X i,h , X i+h,h , i ∈ Z} is an 2h-dependent stationary process, and since X i and X i+h,h are independent, i.e., E X i , X i+h,h = 0 for all i ∈ Z, { X i,h , X i+h,h , i ∈ Z} is then a mean zero 2h-dependent stationary process which implies that n −1/2 n i=1 X i,h , X i+h,h = O P (1).…”

Section: Appendix : Proofsmentioning

confidence: 93%

Moving block and tapered block bootstrap for functional time series with an application to the $K$-sample mean problem

Pilavakis¹,

Paparoditis²,

Sapatinas³

2019

Bernoulli

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We consider infinite-dimensional Hilbert space-valued random variables that are assumed to be temporal dependent in a broad sense. We prove a central limit theorem for the moving block bootstrap and for the tapered block bootstrap, and show that these block bootstrap procedures also provide consistent estimators of the long run covariance operator. Furthermore, we consider block bootstrap-based procedures for fully functional testing of the equality of mean functions between several independent functional time series. We establish validity of the block bootstrap methods in approximating the distribution of the statistic of interest under the null and show consistency of the block bootstrap-based tests under the alternative. The finite sample behaviour of the procedures is investigated by means of simulations. An application to a real-life dataset is also discussed. n )) − nE(X n ⊗ X n ) HS = o P (1), in probability. The Tapered Block BootstrapThe TBB procedure is a modification of the block bootstrap procedure considered in Section 2.2 which is obtained by introducing a tapering of the random elements X t . The tapering function downweights the endpoints of each block B i , towards zero, i.e., towards the mean function of X t . The pseudo observations are then obtained by choosing, with replacement, k appropriately scaled and tapered blocks of length b of centered observations and joining them together.

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“…For weakly dependent observations strong consistency (almost surely convergence) of the X 1 * n and X 2 * n were proved in Shao and Yu [19] and Peligrad [18]. Gonçalves and White [9] and Dehling, Sharipov and Wendler [5] extended these results to functionals of mixing processes. In this paper we will concentrate on bootstrap for U -statistics of weakly dependent observations.…”

Section: Introductionmentioning

confidence: 89%

Bootstrap forU-statistics: a new approach

Sharipov

Tewes

Wendler

2016

Journal of Nonparametric Statistics

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Bootstrap for nonlinear statistics like U-statistics of dependent data has been studied by several authors. This is typically done by producing a bootstrap version of the sample and plugging it into the statistic. We suggest an alternative approach of getting a bootstrap version of U-statistics. We will show the consistency of the new method and compare its finite sample properties in a simulation study and by applying both methods to financial data.

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Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics

Cited by 26 publications

References 32 publications

Sequential block bootstrap in a Hilbert space with application to change point analysis

Sequential block bootstrap in a Hilbert space with application to change point analysis

Moving block and tapered block bootstrap for functional time series with an application to the $K$-sample mean problem

Bootstrap forU-statistics: a new approach

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