## The Annals of Mathematical Statistics

### On the Distribution of the Likelihood Ratio

Robert V. Hogg

#### 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

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