## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 41, Number 1 (1970), 214-226.

### The Weighted Likelihood Ratio, Sharp Hypotheses about Chances, the Order of a Markov Chain

James M. Dickey and B. P. Lientz

#### Abstract

The Bayesian theory for testing a sharp hypothesis, defined by fixed values of parameters, is here presented in general terms. Arbitrary positive prior probability is attached to the hypothesis. The ratio of posterior to prior odds for the hypothesis is given by the weighted likelihood ratio, shown here to equal Leonard J. Savage's (1963) ratio of a posterior to a prior density (2.21). This Bayesian approach to hypothesis testing was suggested by Jeffreys (1948), Savage (1959), (1961), Lindley (1961), and Good (1950), (1965), but obscured some what by approximations and unique choices of prior distributions. This Bayesian theory is distinct from that of Lindley (1965) and that of Dickey (1967a). Applications are given to hypotheses about multinomial means, for example, equality of two binomial probabilities. A new test is presented for the order of a finite-state Markov chain.

#### Article information

**Source**

Ann. Math. Statist., Volume 41, Number 1 (1970), 214-226.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177697203

**Digital Object Identifier**

doi:10.1214/aoms/1177697203

**Mathematical Reviews number (MathSciNet)**

MR258187

**Zentralblatt MATH identifier**

0188.50102

**JSTOR**

links.jstor.org

#### Citation

Dickey, James M.; Lientz, B. P. The Weighted Likelihood Ratio, Sharp Hypotheses about Chances, the Order of a Markov Chain. Ann. Math. Statist. 41 (1970), no. 1, 214--226. doi:10.1214/aoms/1177697203. https://projecteuclid.org/euclid.aoms/1177697203

#### Corrections

- See Correction: James Dickey, B. P. Lientz. Correction Notes: Correction to "The Weighted Likelihood Ratio, Sharp Hypotheses About Chances, the Order of a Markov Chain". Ann. Math. Statist., Volume 43, Number 4 (1972), 1380--1380.Project Euclid: euclid.aoms/1177692496
- See Correction: James Dickey, B. P. Lientz. Correction Notes: Correction to "The Weighted Likelihood Ratio, Sharp Hypotheses About Chances, the Order of a Markov Chain". Ann. Math. Statist., Volume 41, Number 4 (1970), 1374--1374.Project Euclid: euclid.aoms/1177696919