Statistical Inference about Markov Chains

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 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 uth order Markov chain against the alternative it is rth but not uth 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 ²-tests of the form used in contingency tables.