WEAKLY DEPENDENT FUNCTIONAL DATA

doi:10.1214/09-AOS768

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WEAKLY DEPENDENT FUNCTIONAL DATA

2010-06-01

267

Siegfried Hörmann *

Piotr Kokoszka

DOI：10.1214/09-AOS768

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Abstract

En 中文

Functional data often arise from measurements on tine time grids and are obtained by separating an almost continuous time record into natural consecutive intervals, or 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 sigma-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 in-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.

Keywords:

Asymptotics

change points

eigenfunctions

functional principal components

functional time series

long-run variance

weak dependence