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The Bootstrap for the Functional Autoregressive Model FAR(1)

  • Functional data analysis is a branch of statistics that deals with observations \(X_1,..., X_n\) which are curves. We are interested in particular in time series of dependent curves and, specifically, consider the functional autoregressive process of order one (FAR(1)), which is defined as \(X_{n+1}=\Psi(X_{n})+\epsilon_{n+1}\) with independent innovations \(\epsilon_t\). Estimates \(\hat{\Psi}\) for the autoregressive operator \(\Psi\) have been investigated a lot during the last two decades, and their asymptotic properties are well understood. Particularly difficult and different from scalar- or vector-valued autoregressions are the weak convergence properties which also form the basis of the bootstrap theory. Although the asymptotics for \(\hat{\Psi}{(X_{n})}\) are still tractable, they are only useful for large enough samples. In applications, however, frequently only small samples of data are available such that an alternative method for approximating the distribution of \(\hat{\Psi}{(X_{n})}\) is welcome. As a motivation, we discuss a real-data example where we investigate a changepoint detection problem for a stimulus response dataset obtained from the animal physiology group at the Technical University of Kaiserslautern. To get an alternative for asymptotic approximations, we employ the naive or residual-based bootstrap procedure. In this thesis, we prove theoretically and show via simulations that the bootstrap provides asymptotically valid and practically useful approximations of the distributions of certain functions of the data. Such results may be used to calculate approximate confidence bands or critical bounds for tests.

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Metadaten
Verfasserangaben:Euna Gesare Nyarige
URN (Permalink):urn:nbn:de:hbz:386-kluedo-44105
Betreuer:Jürgen Franke
Dokumentart:Dissertation
Sprache der Veröffentlichung:Englisch
Veröffentlichungsdatum (online):07.05.2016
Jahr der Veröffentlichung:2016
Veröffentlichende Institution:Technische Universität Kaiserslautern
Titel verleihende Institution:Technische Universität Kaiserslautern
Datum der Annahme der Abschlussarbeit:22.06.2016
Datum der Publikation (Server):06.07.2016
Freies Schlagwort / Tag:Autoregressive Hilbertian model; Bootstrap; Functional autoregression; Functional time series
Seitenzahl:XVI, 152
Fachbereiche / Organisatorische Einheiten:Fachbereich Mathematik
DDC-Sachgruppen:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Klassifikation (Mathematik):62-XX STATISTICS
Lizenz (Deutsch):Standard gemäß KLUEDO-Leitlinien vom 30.07.2015