Annals of Statistics

Weakly dependent functional data

Siegfried Hörmann and Piotr Kokoszka

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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.

Article information

Ann. Statist., Volume 38, Number 3 (2010), 1845-1884.

First available in Project Euclid: 24 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 60G10: Stationary processes 62G05: Estimation

Asymptotics change points eigenfunctions functional principal components functional time series long-run variance weak dependence


Hörmann, Siegfried; Kokoszka, Piotr. Weakly dependent functional data. Ann. Statist. 38 (2010), no. 3, 1845--1884. doi:10.1214/09-AOS768.

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