Reaction times of monitoring schemes for ARMA time series

This paper is concerned with deriving the limit distributions of stopping times devised to sequentially uncover structural breaks in the parameters of an autoregressive moving average, ARMA, time series. The stopping rules are defined as the first time lag for which detectors, based on CUSUMs and Page's CUSUMs for residuals, exceed the value of a prescribed threshold function. It is shown that the limit distributions crucially depend on a drift term induced by the underlying ARMA parameters. The precise form of the asymptotic is determined by an interplay between the location of the break point and the size of the change implied by the drift. The theoretical results are accompanied by a simulation study and applications to electroencephalography, EEG, and IBM data. The empirical results indicate a satisfactory behavior in finite samples.Comment: Published at http://dx.doi.org/10.3150/14-BEJ604 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Functional Data Analysis with Increasing Number of Projections

Functional principal components (FPC's) provide the most important and most extensively used tool for dimension reduction and inference for functional data. The selection of the number, d, of the FPC's to be used in a specific procedure has attracted a fair amount of attention, and a number of reasonably effective approaches exist. Intuitively, they assume that the functional data can be sufficiently well approximated by a projection onto a finite-dimensional subspace, and the error resulting from such an approximation does not impact the conclusions. This has been shown to be a very effective approach, but it is desirable to understand the behavior of many inferential procedures by considering the projections on subspaces spanned by an increasing number of the FPC's. Such an approach reflects more fully the infinite-dimensional nature of functional data, and allows to derive procedures which are fairly insensitive to the selection of d. This is accomplished by considering limits as d tends to infinity with the sample size. We propose a specific framework in which we let d tend to infinity by deriving a normal approximation for the two-parameter partial sum process of the scores \xi_{i,j} of the i-th function with respect to the j-th FPC. Our approximation can be used to derive statistics that use segments of observations and segments of the FPC's. We apply our general results to derive two inferential procedures for the mean function: a change-point test and a two-sample test. In addition to the asymptotic theory, the tests are assessed through a small simulation study and a data example

Changepoint-Analyse für Kenngrößen der Telekommunikation

Thema der Dissertation ist die theoretische Untersuchung von Verfahren der Changepoint-Analyse zur Anwendung auf Telekommunikationsdaten. Die hier betrachteten Modelle sind durch Modelle motiviert, wie sie für Fragestellungen der Telekommunikation verwendet werden. Insbesondere für Marktanteilsuntersuchungen bezogen auf telefonierte Minuten erweisen sich lineare Modelle als geeignet. Die Fehlerterme sind dabei in der Praxis häufig nicht unabhängig, wie in theoretischen Untersuchungen oft vorausgesetzt wird, sondern als korreliert anzusehen. Aus dieser Motivation heraus verallgemeinern wir Verfahren der a posteriori Changepoint-Analyse für lineare Modelle mit unabhängigen Fehlern auf solche mit korrelierten Fehlertermen. Eine weitere wichtige sehr allgemein definierte Klasse von Modellen ist die Modellklasse der State-Space Modelle. State-Space Modelle werden insbesondere für Prognosen des Verkehrsaufkommens im Telekommunikationsbereich herangezogen. Es werden neue Verfahren zur a posteriori Changepoint-Analyse für diese Modellklasse entwickelt und auf ihre asymptotischen Eigenschaften untersucht. Bezüglich sequentieller Verfahren der Changepoint-Analyse stellen wir bekannte praxisorientierte Verfahren vor und setzen diese auf die hier betrachteten linearen Modelle sowie State-Space Modelle um. Theoretisch erzielte Ergebnisse werden durch Simulationen überprüft. Sämtliche Programme, die zu Simulationsstudien verwendet werden, sind in der statistischen Programmiersprache R geschrieben. Die hier untersuchten Verfahren werden in einer Anwendungsstudie, die aus Datenschutzgründen nicht Bestandteil der Dissertation ist, auf Realdaten der Deutschen Telekom angewendet. Dabei wird einerseits die Konzeption von Frühwarnsystemen diskutiert, die sequentiell Beobachtungen auf Strukturbrüche (Abweichungen von einem vorgegebenen Modell) untersuchen. Andererseits betrachten wir dort Analysesysteme, die eine Menge von Beobachtungen im Nachhinein (''a posteriori'') auf vorhandene Changepoints überprüfen. Die Verfahren liefern in praktischen Anwendungen wertvolle Hinweise auf Strukturbrüche in beobachteten Daten. Ein Teil der Verfahren wird von der Deutschen Telekom zur automatisierten Überwachung von Marktanteilen sowie anderen Zielgrößen verwendet

Reaction times of monitoring schemes for ARMA time series

This paper is concerned with deriving the limit distributions of stopping times devised to sequentially uncover structural breaks in the parameters of an autoregressive moving average, ARMA, time series. The stopping rules are defined as the first time lag for which detectors, based on CUSUMs and Page's CUSUMs for residuals, exceed the value of a prescribed threshold function. It is shown that the limit distributions crucially depend on a drift term induced by the underlying ARMA parameters. The precise form of the asymptotic is determined by an interplay between the location of the break point and the size of the change implied by the drift. The theoretical results are accompanied by a simulation study and applications to electroencephalography, EEG, and IBM data. The empirical results indicate a satisfactory behavior in finite samples

Invariance principles for renewal processes when only moments of low order exist

AbstractStarting from recent strong and weak approximations to the partial sums of i.i.d. random vectors (cf. U. Einmahl, Ann. Probab., 15 1419–1440), some corresponding invariance principles are developed for associated renewal processes and random sums. Optimality of the approximation is proved in the case when only two moments exist. Among other applications, a Darling-Erdös type extreme value theorem for renewal processes will be derived

Invariance principles for renewal processes when only moments of low order exist

Starting from recent strong and weak approximations to the partial sums of i.i.d. random vectors (cf. U. Einmahl, Ann. Probab., 15 1419-1440), some corresponding invariance principles are developed for associated renewal processes and random sums. Optimality of the approximation is proved in the case when only two moments exist. Among other applications, a Darling-Erdös type extreme value theorem for renewal processes will be derived.Invariance principles strong approximations weak approximations renewal processes random sums Wiener process extreme value theorem

On the optimality of strong approximation rates for compound renewal processes

The optimality of certain approximation rates appearing in strong invariance principles for partial sums indexed by a renewal process is discussed. The results extend and unify earlier work on the best rates in the invariance principles for renewal counting processes. The motivation for this note came from a recent approximation of compound renewal processes due to Csörgo, Deheuvels and Horváth (1987), which is presented here in a slightlty extended version.compound renewal processes Wiener process strong invariance principles optimal approximation rates