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Determination of the order of a Markov chain by Akaike's information criterion

Published online by Cambridge University Press:  14 July 2016

H. Tong
H. Tong*
Affiliation:
Institute of Statistical Mathematics, Tokyo
*
*At present at University of Manchester Institute of Science and Technology, Manchester, England.
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Abstract

Using Akaike's information criterion, we have presented an objective procedure for the determination of the order of an ergodic Markov chain with a finite number of states. The procedure exploits the asymptotic properties of the maximum likelihood ratio statistics and Kullback and Leibler's mean information for the discrimination between two distributions. Numerical illustrations are given, using data from Bartlett (1966), Good and Gover (1967) and some weather records.

Keywords

MARKOV CHAINS OF ORDER kAKAIKE'S INFORMATION CRITERIONKULLBACK-LEIBLER'S INFORMATIONNORMAL REAL NUMBER
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 3 , September 1975 , pp. 488 - 497
Copyright
Copyright © Applied Probability Trust 1975 

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References

Akaike, H. (1970) Statistical predictor identification. Ann. Inst. Statist. Math. 22, 203–207.CrossRefGoogle Scholar
Akaike, H. (1972) Information theory and an extension of the maximum likelihood principle. Second International Symposium on Information Theory , Ed. Petrov, B. N. and Csáki, F. Akadémiai Kiadó, Budapest 267–281.Google Scholar
Akaike, H. (1974) A new look at the statistical identification model. IEEE. Trans. Auto. Control AC–19, 716–723.Google Scholar
Anderson, T. W. and Goodman, L. A. (1957) Statistical inference about Markov chains. Ann. Math. Statist. 28, 89–110.Google Scholar
Bartlett, M. S. (1951) The frequency goodness of fit tests for probability chains. Proc. Camb. Phil. Soc. 47, 86–95.CrossRefGoogle Scholar
Bartlett, M. S. (1966) An Introduction to Stochastic Processes with Special Reference to Methods and Applications. Cambridge University Press.Google Scholar
Billingsley, P. (1961) Statistical Inference for Markov Processes. Holt, New York.Google Scholar
Brooks, C. E. P. and Carruthers, N. C. (1953) Handbook of Statistical Methods in Meteorology , H.M.S.O., London.Google Scholar
Gabriel, K. R. and Neumann, J. (1962) A Markov chain model for daily rainfall occurrence in Tel Aviv. Quart. J. R. Met. Soc. 88, 90–95.CrossRefGoogle Scholar
Gani, J. (1955) Some theorems and sufficiency conditions for the maximum likelihood estimator of an unknown parameter in a simple Markov chain. Biometrika 42, 342–359.Google Scholar
Good, I. J. (1953) The serial test for sampling numbers and other tests for randomness. Proc. Camb. Phil. Soc. 49, 276–284.Google Scholar
Good, I. J. and Gover, T. N. (1967) The generalized serial test and the binary expansion of √2. J. R. Statist. Soc. A 130, 102–107.Google Scholar
Hoel, P. G. (1954) A test for Markov chains. Biometrika 41, 430–433.CrossRefGoogle Scholar
Jones, R. H. (1974) Identification and autoregression spectrum estimation. IEEE. Trans. Auto. Control AC–19, 894–898.Google Scholar
Kennedy, W. J. and Bancroft, T. A. (1971) Model building for prediction in regression based upon repeated significance tests. Ann. Math. Statist. 42, 1273–1284.Google Scholar
Kullback, S. (1959) Information Theory and Statistics. John Wiley, New York.Google Scholar
Otoma, T., Nakagawa, T. and Akaike, H. (1972) Statistical approach to computer control of cement rotary kiln. Automatica 8, 35–48.Google Scholar
Parzen, E. (1974) Some solutions to the time series modeling and prediction problem. IEEE. Trans. Auto. Control AC–19, 723–730.CrossRefGoogle Scholar
Tanabe, K. (1974) Statistical regularization of a noisy ill-conditioned system of linear equations by Akaike's information criterion. Research Memo. No. 60, The Institute of Statistical Mathematics, Tokyo, Japan.Google Scholar