Weak convergence of the conditional U-statistics for locally stationary functional time series

In recent years, the direction has turned to non-stationary time series. Here the situation is more complicated: it is often unclear how to set down a meaningful asymptotic for non-stationary processes. For this reason, the theory of locally stationary processes arose, and it is based on infill asym...

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Vydané v:	Statistical inference for stochastic processes : an international journal devoted to time series analysis and the statistics of continuous time processes and dynamic systems Ročník 27; číslo 2; s. 227 - 304
Hlavní autori:	Soukarieh, Inass, Bouzebda, Salim
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Dordrecht Springer Netherlands 01.07.2024 Springer Nature B.V Springer Verlag
Predmet:	Cluster analysis Clustering Convergence Data analysis Functional Analysis Functionals Mathematics Mathematics and Statistics Operators (mathematics) Probability Probability Theory and Stochastic Processes Random variables Redevelopment Stationary processes Statistical Theory and Methods Statistics Statistics Theory Time series 62G05 processes 62G07 62G08 Secondary 62E20 Regression Conditional VC-class of functions Empirical processes Kernel-type estimators Locally stationary functional 62G30 62G20 62G32 62G35 statistics Conditional U-statistics regression Locally Stationary Functional Conditional U-processes
ISSN:	1387-0874, 1572-9311
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Abstract	In recent years, the direction has turned to non-stationary time series. Here the situation is more complicated: it is often unclear how to set down a meaningful asymptotic for non-stationary processes. For this reason, the theory of locally stationary processes arose, and it is based on infill asymptotics created from non-parametric statistics. The present paper aims to develop a framework for inference of locally stationary functional time series based on the so-called conditional U -statistics introduced by Stute (Ann Probab 19:812–825, 1991), and may be viewed as a generalization of the Nadaraya-Watson regression function estimates. In this paper, we introduce an estimator of the conditional U -statistics operator that takes into account the nonstationary behavior of the data-generating process. We are mainly interested in establishing weak convergence of conditional U -processes in the locally stationary functional mixing data framework. More precisely, we investigate the weak convergence of conditional U -processes when the explicative variable is functional. We treat the weak convergence when the class of functions is bounded or unbounded, satisfying some moment conditions. These results are established under fairly general structural conditions on the classes of functions and the underlying models. The theoretical results established in this paper are (or will be) critical tools for further functional data analysis developments.
AbstractList	In recent years, the direction has turned to non-stationary time series. Here the situation is more complicated: it is often unclear how to set down a meaningful asymptotic for non-stationary processes. For this reason, the theory of locally stationary processes arose, and it is based on infill asymptotics created from non-parametric statistics. The present paper aims to develop a framework for inference of locally stationary functional time series based on the so-called conditional U -statistics introduced by Stute (Ann Probab 19:812–825, 1991), and may be viewed as a generalization of the Nadaraya-Watson regression function estimates. In this paper, we introduce an estimator of the conditional U -statistics operator that takes into account the nonstationary behavior of the data-generating process. We are mainly interested in establishing weak convergence of conditional U -processes in the locally stationary functional mixing data framework. More precisely, we investigate the weak convergence of conditional U -processes when the explicative variable is functional. We treat the weak convergence when the class of functions is bounded or unbounded, satisfying some moment conditions. These results are established under fairly general structural conditions on the classes of functions and the underlying models. The theoretical results established in this paper are (or will be) critical tools for further functional data analysis developments. In recent years, the direction has turned to non-stationary time series. Here the situation is more complicated: it is often unclear how to set down a meaningful asymptotic for non-stationary processes. For this reason, the theory of locally stationary processes arose, and it is based on infill asymptotics created from non-parametric statistics. The present paper aims to develop a framework for inference of locally stationary functional time series based on the so-called conditional $U$-statistics introduced by W. Stute [Ann. Probab. 19 (1991) 812--825], and may be viewed as a generalization of the Nadaraya-Watson regression function estimates. In this paper, we introduce an estimator of the conditional $U$-statistics operator that takes into account the nonstationary behavior of the data-generating process. We are mainly interested in establishing weak convergence of conditional $U$-processes in the locally stationary functional mixing data framework. More precisely, we investigate the weak convergence of conditional $U$-processes when the explicative variable is functional. We treat the weak convergence when the class of functions is bounded or unbounded, satisfying some moment conditions. These results are established under fairly general structural conditions on the classes of functions and the underlying models.The theoretical results established in this paper are (or will be) critical tools for further functional data analysis developments. In recent years, the direction has turned to non-stationary time series. Here the situation is more complicated: it is often unclear how to set down a meaningful asymptotic for non-stationary processes. For this reason, the theory of locally stationary processes arose, and it is based on infill asymptotics created from non-parametric statistics. The present paper aims to develop a framework for inference of locally stationary functional time series based on the so-called conditional U-statistics introduced by Stute (Ann Probab 19:812–825, 1991), and may be viewed as a generalization of the Nadaraya-Watson regression function estimates. In this paper, we introduce an estimator of the conditional U-statistics operator that takes into account the nonstationary behavior of the data-generating process. We are mainly interested in establishing weak convergence of conditional U-processes in the locally stationary functional mixing data framework. More precisely, we investigate the weak convergence of conditional U-processes when the explicative variable is functional. We treat the weak convergence when the class of functions is bounded or unbounded, satisfying some moment conditions. These results are established under fairly general structural conditions on the classes of functions and the underlying models. The theoretical results established in this paper are (or will be) critical tools for further functional data analysis developments.
Author	Bouzebda, Salim Soukarieh, Inass
Author_xml	– sequence: 1 givenname: Inass surname: Soukarieh fullname: Soukarieh, Inass organization: LMAC (Laboratory of Applied Mathematics of Compiègne), Université de Technologie de Compiègne – sequence: 2 givenname: Salim orcidid: 0000-0001-7801-4945 surname: Bouzebda fullname: Bouzebda, Salim email: salim.bouzebda@utc.fr organization: LMAC (Laboratory of Applied Mathematics of Compiègne), Université de Technologie de Compiègne
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Keywords	62G05 processes 62G07 62G08 Secondary 62E20 Regression Conditional VC-class of functions Empirical processes Kernel-type estimators Locally stationary functional 62G30 62G20 62G32 62G35 statistics Conditional U-statistics regression Locally Stationary Functional Conditional U-processes
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Snippet	In recent years, the direction has turned to non-stationary time series. Here the situation is more complicated: it is often unclear how to set down a...
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SubjectTerms	Cluster analysis Clustering Convergence Data analysis Functional Analysis Functionals Mathematics Mathematics and Statistics Operators (mathematics) Probability Probability Theory and Stochastic Processes Random variables Redevelopment Stationary processes Statistical Theory and Methods Statistics Statistics Theory Time series
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Title	Weak convergence of the conditional U-statistics for locally stationary functional time series
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