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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Published in:Statistical inference for stochastic processes : an international journal devoted to time series analysis and the statistics of continuous time processes and dynamic systems Vol. 27; no. 2; pp. 227 - 304
Main Authors: Soukarieh, Inass, Bouzebda, Salim
Format: Journal Article
Language:English
Published: Dordrecht Springer Netherlands 01.07.2024
Springer Nature B.V
Springer Verlag
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:1387-0874
1572-9311
DOI:10.1007/s11203-023-09305-y