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Generalized adaptive exponential smoothing of observations from an ergodic hidden Markov model

Part of: Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Ulrich Herkenrath
Ulrich Herkenrath*
Affiliation:
Gerhard-Mercator-Universität Duisburg
*
∗Postal address: Fachbereich 11 Mathematik, Gerhard-Mercator-Universität Duisburg, Postfach 10 15 03, D-47048 Duisburg, Germany. Email address: herkenrath@math.uni-duisburg.de.
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Abstract

We consider a sequence of observations which is generated by a so-called hidden Markov model. An exponential smoothing procedure applied to such an observation sequence generates an inhomogeneous Markov process as a sequence of smoothed values. If the state sequence of the underlying hidden Markov model is moreover ergodic, then for two classes of smoothing functions the strong ergodicity of the sequence of smoothed values is proved. As a consequence a central limit theorem and a law of large numbers hold true for the smoothed values. The proof uses general results for so-called convergent inhomogeneous Markov processes. The procedure proposed by the author can be applied to some time series discussed in the literature.

Keywords

Smoothing of non-linear time seriesgeneralized exponential smoothinghidden Markov modelsconvergent inhomogeneous Markov processes

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 987 - 998
Copyright
Copyright © Applied Probability Trust 1999 

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