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March, 1957 Statistical Inference about Markov Chains
T. W. Anderson, Leo A. Goodman
Ann. Math. Statist. 28(1): 89-110 (March, 1957). DOI: 10.1214/aoms/1177707039

Abstract

Maximum likelihood estimates and their asymptotic distribution are obtained for the transition probabilities in a Markov chain of arbitrary order when there are repeated observations of the chain. Likelihood ratio tests and $\chi^2$-tests of the form used in contingency tables are obtained for testing the following hypotheses: (a) that the transition probabilities of a first order chain are constant, (b) that in case the transition probabilities are constant, they are specified numbers, and (c) that the process is a $u$th order Markov chain against the alternative it is $r$th but not $u$th order. In case $u = 0$ and $r = 1$, case (c) results in tests of the null hypothesis that observations at successive time points are statistically independent against the alternate hypothesis that observations are from a first order Markov chain. Tests of several other hypotheses are also considered. The statistical analysis in the case of a single observation of a long chain is also discussed. There is some discussion of the relation between likelihood ratio criteria and $\chi^2$-tests of the form used in contingency tables.

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T. W. Anderson. Leo A. Goodman. "Statistical Inference about Markov Chains." Ann. Math. Statist. 28 (1) 89 - 110, March, 1957. https://doi.org/10.1214/aoms/1177707039

Information

Published: March, 1957
First available in Project Euclid: 27 April 2007

zbMATH: 0087.14905
MathSciNet: MR84903
Digital Object Identifier: 10.1214/aoms/1177707039

Rights: Copyright © 1957 Institute of Mathematical Statistics

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Vol.28 • No. 1 • March, 1957
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