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Mixture models for time series

Part of: Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Assad Jalali  and
John Pemberton
Assad Jalali
Affiliation:
University of Swansea
John Pemberton*
Affiliation:
University of Swansea
*
∗Postal address for both authors: Statistical and Operational Research Group, European Business Management School, University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, UK.
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Abstract

In this paper we extend the class of zero-order threshold autoregressive models to a much richer class of mixture models. The new class has the important property of duality which, as we show, corresponds to time reversal. We are then able to obtain the time reversals of the zero-order threshold models and to characterise the time-reversible members of this subclass. These turn out to be quite trivial. The time-reversible models of the more general class do not suffer in this way. The complete stationary distributional structure is given, as are various moments, in particular the autocovariance function. This is shown to be of ARMA type. Finally we give two examples, the second of which extends from the finite to the countable mixture case. The general theory for this extension will be given elsewhere.

Keywords

ARMA COVARIANCESCONDITIONAL DISTRIBUTIONSMARKOV CHAINMIXTURE DISTRIBUTIONNON-LINEAR TIME SERIES MODELSTATIONARY DISTRIBUTIONSSTOCHASTIC MATRIXTIME REVERSALZERO-ORDER THRESHOLD AUTOREGRESSION

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 123 - 138
Copyright
Copyright © Applied Probability Trust 1995 

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