Non-Parametric Conditional U-Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design

Stute presented the so-called conditional U-statistics generalizing the Nadaraya–Watson estimates of the regression function. Stute demonstrated their pointwise consistency and the asymptotic normality. In this paper, we extend the results to a more abstract setting. We develop an asymptotic theory...

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Bibliographic Details
Published in:Mathematics Vol. 11; no. 1; p. 16
Main Authors: Bouzebda, Salim, Soukarieh, Inass
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.01.2023
MDPI
Subjects:
[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA]
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
Asymptotic properties
conditional U-processes
conditional U-statistics
conditional U-statistics; locally stationary random field; functional data; empirical processes; conditional U-processes; VC-class of functions; kernel-type estimators; regression; irregularly spaced data
Convergence
Data analysis
empirical processes
Fields (mathematics)
Functional Analysis
functional data
Functionals
locally stationary random field
Mathematics
Nonparametric statistics
Normality
Probability
QA1-939
Random variables
Sampling designs
Spatial data
Statistics
Statistics Theory
Stochastic processes
VC-class of functions
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Description
Summary:Stute presented the so-called conditional U-statistics generalizing the Nadaraya–Watson estimates of the regression function. Stute demonstrated their pointwise consistency and the asymptotic normality. In this paper, we extend the results to a more abstract setting. We develop an asymptotic theory of conditional U-statistics for locally stationary random fields {Xs,An:sinRn} observed at irregularly spaced locations in Rn=[0,An]d as a subset of Rd. We employ a stochastic sampling scheme that may create irregularly spaced sampling sites in a flexible manner and includes both pure and mixed increasing domain frameworks. We specifically examine the rate of the strong uniform convergence and the weak convergence of conditional U-processes when the explicative variable is functional. We examine the weak convergence where the class of functions is either bounded or unbounded and satisfies specific moment conditions. These results are achieved under somewhat general structural conditions pertaining to the classes of functions and the underlying models. The theoretical results developed in this paper are (or will be) essential building blocks for several future breakthroughs in functional data analysis.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math11010016