Locally stationary functional time series

Anne van Delft*, Michael Eichler

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

The literature on time series of functional data has focused on processes of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be able to weaken this assumption. This paper introduces a framework that will enable meaningful statistical inference of functional data of which the dynamics change over time. We put forward the concept of local stationarity in the functional setting and establish a class of processes that have a functional time-varying spectral representation. Subsequently, we derive conditions that allow for fundamental results from nonstationary multivariate time series to carry over to the function space. In particular, time-varying functional ARMA processes are investigated and shown to be functional locally stationary according to the proposed definition. As a side-result, we establish a Cramer representation for an important class of weakly stationary functional processes. Important in our context is the notion of a time-varying spectral density operator of which the properties are studied and uniqueness is derived. Finally, we provide a consistent nonparametric estimator of this operator and show it is asymptotically Gaussian using a weaker tightness criterion than what is usually deemed necessary.
Original languageEnglish
Pages (from-to)107-170
Number of pages64
JournalElectronic Journal of Statistics
Volume12
Issue number1
DOIs
Publication statusPublished - 2018

Keywords

  • Functional data analysis
  • locally stationary processes
  • spectral analysis
  • kernel estimator
  • PRINCIPAL COMPONENT ANALYSIS
  • NONSTATIONARY PROCESSES
  • ASYMPTOTIC THEORY
  • YIELD CURVE
  • DATA MODELS
  • REGRESSION
  • MORTALITY
  • FERTILITY

Cite this

van Delft, Anne ; Eichler, Michael. / Locally stationary functional time series. In: Electronic Journal of Statistics. 2018 ; Vol. 12, No. 1. pp. 107-170.
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abstract = "The literature on time series of functional data has focused on processes of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be able to weaken this assumption. This paper introduces a framework that will enable meaningful statistical inference of functional data of which the dynamics change over time. We put forward the concept of local stationarity in the functional setting and establish a class of processes that have a functional time-varying spectral representation. Subsequently, we derive conditions that allow for fundamental results from nonstationary multivariate time series to carry over to the function space. In particular, time-varying functional ARMA processes are investigated and shown to be functional locally stationary according to the proposed definition. As a side-result, we establish a Cramer representation for an important class of weakly stationary functional processes. Important in our context is the notion of a time-varying spectral density operator of which the properties are studied and uniqueness is derived. Finally, we provide a consistent nonparametric estimator of this operator and show it is asymptotically Gaussian using a weaker tightness criterion than what is usually deemed necessary.",
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Locally stationary functional time series. / van Delft, Anne; Eichler, Michael.

In: Electronic Journal of Statistics, Vol. 12, No. 1, 2018, p. 107-170.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - van Delft, Anne

AU - Eichler, Michael

N1 - data source: no data used

PY - 2018

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AB - The literature on time series of functional data has focused on processes of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be able to weaken this assumption. This paper introduces a framework that will enable meaningful statistical inference of functional data of which the dynamics change over time. We put forward the concept of local stationarity in the functional setting and establish a class of processes that have a functional time-varying spectral representation. Subsequently, we derive conditions that allow for fundamental results from nonstationary multivariate time series to carry over to the function space. In particular, time-varying functional ARMA processes are investigated and shown to be functional locally stationary according to the proposed definition. As a side-result, we establish a Cramer representation for an important class of weakly stationary functional processes. Important in our context is the notion of a time-varying spectral density operator of which the properties are studied and uniqueness is derived. Finally, we provide a consistent nonparametric estimator of this operator and show it is asymptotically Gaussian using a weaker tightness criterion than what is usually deemed necessary.

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