### Abstract

Language | English |
---|---|

Pages | 1845-1884 |

Number of pages | 40 |

Journal | The annals of statistics |

Volume | 38 |

Issue number | 3 |

Status | Published - 2010 |

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### Cite this

*The annals of statistics*,

*38*(3), 1845-1884.

**Weakly dependent functional data.** / Hörmann, Siegfried; Kokoszka, P.

Research output: Contribution to journal › Article › Research › peer-review

*The annals of statistics*, vol. 38, no. 3, pp. 1845-1884.

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TY - JOUR

T1 - Weakly dependent functional data

AU - Hörmann, Siegfried

AU - Kokoszka, P.

N1 - Language of publication: en

PY - 2010

Y1 - 2010

N2 - 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 Ïƒ-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.

AB - 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 Ïƒ-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.

M3 - Article

VL - 38

SP - 1845

EP - 1884

JO - The annals of statistics

T2 - The annals of statistics

JF - The annals of statistics

SN - 0090-5364

IS - 3

ER -