Change Point Detection with Stable AR(1) Errors
In this paper we develop two types of tests to detect changes in the location parameters of dependent observations with infinite variances. We consider the case of autoregressive processes of order one with independent innovations in the domain of attraction of a stable law. If the d largest (in magnitude) observations are removed from the sample, then the standard CUSUM process developed for weakly dependent observations with finite variance can be used assuming that \(d = d(n) \rightarrow \infty\) and d(n)∕n → 0 as n, the sample size, tends to \(\infty\). We study two types of statistics. In case of the maximally selected CUSUM process we estimate the long run variance by kernel estimators and we develop the corresponding change point test. We also propose ratio statistics which do not depend on the long run variances. Monte Carlo simulations illustrate that the limit results can be used even in case of small and moderate sample sizes.
KeywordsAutoregressive Process Brownian Bridge Empirical Power Change Point Detection Moderate Sample Size
Research of Alina Bazarova supported by Austrian Science Fund (FWF), Projekt W1230-N13. Research of István Berkes supported by FWF grant P24302-N18 and OTKA grant K 108615. Research of Lajor Horváth supported by NSF grant DMS 1305858.
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