## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 27, Number 2 (1956), 529-532.

### On the Distribution of the Likelihood Ratio

#### Abstract

In an investigation of the distribution of the likelihood ratio $\lambda$, Wilks [3] proved, under certain regularity conditions, that $-2 \ln \lambda$ is, except for terms of order $1/\sqrt n$, distributed like $\chi^2$ with $k - m$ degrees of freedom, where $k$ is the dimension of the parameter space $\Omega$ of admissible hypotheses and $m$ is the dimension of the parameter space $\omega$ of null hypotheses. In this paper, we consider the nonregular densities investigated by R. C. Davis [1] and show that for certain hypotheses $-2 \ln \lambda$ has an exact $\chi^2$-distribution with $2(k - m)$ degrees of freedom.

#### Article information

**Source**

Ann. Math. Statist., Volume 27, Number 2 (1956), 529-532.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177728276

**Digital Object Identifier**

doi:10.1214/aoms/1177728276

**Mathematical Reviews number (MathSciNet)**

MR79382

**Zentralblatt MATH identifier**

0071.13303

**JSTOR**

links.jstor.org

#### Citation

Hogg, Robert V. On the Distribution of the Likelihood Ratio. Ann. Math. Statist. 27 (1956), no. 2, 529--532. doi:10.1214/aoms/1177728276. https://projecteuclid.org/euclid.aoms/1177728276