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Frequency and phase estimation in time series with quasi periodic components [Elektronische Ressource] / vorgelegt von Konstantinos Paraschakis

108 pages
INAUGURAL - DISSERTATIONzurErlangung der DoktorwurdederNaturwissenschaftlich-Mathematischen Gesamtfakult atderRuprecht - Karls - Universit atHeidelbergvorgelegt vonM.Sc. StatistikerKonstantinos ParaschakisausIlion (Nea Liosia) / GriechenlandTag der mundlic hen Prufung: 24. Mai 2011iiiiiThema:Frequency and phase estimation in time series with quasi periodiccomponentsGutachter: Prof. Dr. Rainer DahlhausProf. Dr. Mark PodolskijivvIn the memory of my fatherviAcknowledgmentsI would like to thank my family, Antonis, Alexandra and Kalliopi for their support andencouragement. I express my gratitude to Prof. Dr. Rainer Dahlhaus for giving methe opportunity to come to Heidelberg and work on my PhD at a high level Universityand for his great contribution to this project. I extend my thanks to Prof. EvangelosIoannidis for his guidance and valuable advice throughout my studies. I also wish tothank Lilian Villar n for her feedback on computational issues and her support and to Dr.Jan Neddermeyer, Dr. Markus Fischer, Prof. Dr. Jan Johannes, Dr. Cornelia Wichelhausand Dr. Angelika Rohde for their valuable discussions on this work.This work was supported partially by the Deutsche Forschungsgemeinschaft, by theUniversity of Heidelberg and by the Heidelberg international Graduiertenkolleg and Grad-uate School.
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INAUGURAL - DISSERTATION
zur
Erlangung der Doktorwurde
der
Naturwissenschaftlich-Mathematischen Gesamtfakult at
der
Ruprecht - Karls - Universit at
Heidelberg
vorgelegt von
M.Sc. Statistiker
Konstantinos Paraschakis
aus
Ilion (Nea Liosia) / Griechenland
Tag der mundlic hen Prufung: 24. Mai 2011iiiii
Thema:
Frequency and phase estimation in time series with quasi periodic
components
Gutachter: Prof. Dr. Rainer Dahlhaus
Prof. Dr. Mark Podolskijivv
In the memory of my fatherviAcknowledgments
I would like to thank my family, Antonis, Alexandra and Kalliopi for their support and
encouragement. I express my gratitude to Prof. Dr. Rainer Dahlhaus for giving me
the opportunity to come to Heidelberg and work on my PhD at a high level University
and for his great contribution to this project. I extend my thanks to Prof. Evangelos
Ioannidis for his guidance and valuable advice throughout my studies. I also wish to
thank Lilian Villar n for her feedback on computational issues and her support and to Dr.
Jan Neddermeyer, Dr. Markus Fischer, Prof. Dr. Jan Johannes, Dr. Cornelia Wichelhaus
and Dr. Angelika Rohde for their valuable discussions on this work.
This work was supported partially by the Deutsche Forschungsgemeinschaft, by the
University of Heidelberg and by the Heidelberg international Graduiertenkolleg and Grad-
uate School.
viiviii ACKNOWLEDGMENTSAbstract
A classical model in time series analysis is a stationary process superposed by one or
several deterministic sinusoidal components. Di erent methods are applied to estimate the
frequency (!) of those components such as Least Squares Estimation and the maximization
of the periodogram.
In many applications the assumption of a constant frequency is violated and we turn
to a time dependent frequency function (!(s)). For example in the physics literature this
is viewed as nonlinearity of the phase of a process. A way to estimate !(s) is the local
application of the above methods.
In this dissertation we study the maximum periodogram method on data segments as
an estimator of!(s) and subsequently a least squares technique for estimating the phase.
We prove consistency and asymptotic normality in the context of \in ll asymptotics", a
concept that o ers a meaningful asymptotic theory in cases of local estimations. Finally,
we investigate an estimator based on a local linear approximation of the frequency function,
prove its consistency and asymptotic normality in the \in ll asymptotics" sense and show
that it delivers better estimations than the ordinary periodogram. The theoretical results
are also supported by some simulations.
ixx ABSTRACT

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