Open Access
June 2010 Weakly dependent functional data
Siegfried Hörmann, Piotr Kokoszka
Ann. Statist. 38(3): 1845-1884 (June 2010). DOI: 10.1214/09-AOS768


Functional data often arise from measurements on fine time grids and are obtained by separating an almost continuous time record into natural consecutive intervals, for example, days. The functions thus obtained form a functional time series, and the central issue in the analysis of such data consists in taking into account the temporal dependence of these functional observations. Examples include daily curves of financial transaction data and daily patterns of geophysical and environmental data. For scalar and vector valued stochastic processes, a large number of dependence notions have been proposed, mostly involving mixing type distances between σ-algebras. In time series analysis, measures of dependence based on moments have proven most useful (autocovariances and cumulants). We introduce a moment-based notion of dependence for functional time series which involves m-dependence. We show that it is applicable to linear as well as nonlinear functional time series. Then we investigate the impact of dependence thus quantified on several important statistical procedures for functional data. We study the estimation of the functional principal components, the long-run covariance matrix, change point detection and the functional linear model. We explain when temporal dependence affects the results obtained for i.i.d. functional observations and when these results are robust to weak dependence.


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Siegfried Hörmann. Piotr Kokoszka. "Weakly dependent functional data." Ann. Statist. 38 (3) 1845 - 1884, June 2010.


Published: June 2010
First available in Project Euclid: 24 March 2010

zbMATH: 1189.62141
MathSciNet: MR2662361
Digital Object Identifier: 10.1214/09-AOS768

Primary: 62M10
Secondary: 60G10 , 62G05

Keywords: asymptotics , Change points , Eigenfunctions , functional principal components , functional time series , long-run variance , Weak dependence

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 3 • June 2010
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