Reaction times of monitoring schemes for ARMA time series

Abstract

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.