## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 28, Number 1 (1957), 89-110.

### Statistical Inference about Markov Chains

T. W. Anderson and Leo A. Goodman

#### 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.

#### Article information

**Source**

Ann. Math. Statist., Volume 28, Number 1 (1957), 89-110.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177707039

**Digital Object Identifier**

doi:10.1214/aoms/1177707039

**Mathematical Reviews number (MathSciNet)**

MR84903

**Zentralblatt MATH identifier**

0087.14905

**JSTOR**

links.jstor.org

#### Citation

Anderson, T. W.; Goodman, Leo A. Statistical Inference about Markov Chains. Ann. Math. Statist. 28 (1957), no. 1, 89--110. doi:10.1214/aoms/1177707039. https://projecteuclid.org/euclid.aoms/1177707039