On the Distribution of the Likelihood Ratio

Abstract

A classical result due to Wilks [1] on the distribution of the likelihood ratio $\lambda$ is the following. Under suitable regularity conditions, if the hypothesis that a parameter $\theta$ lies on an $r$-dimensional hyperplane of $k$-dimensional space is true, the distribution of $-2 \log \lambda$ is asymptotically that of $\chi^2$ with $k - r$ degrees of freedom. In many important problems it is desired to test hypotheses which are not quite of the above type. For example, one may wish to test whether $\theta$ is on one side of a hyperplane, or to test whether $\theta$ is in the positive quadrant of a two-dimensional space. The asymptotic distribution of $-2 \log \lambda$ is examined when the value of the parameter is a boundary point of both the set of $\theta$ corresponding to the hypothesis and the set of $\theta$ corresponding to the alternative. First the case of a single observation from a multivariate normal distribution, with mean $\theta$ and known covariance matrix, is treated. The general case is then shown to reduce to this special case where the covariance matrix is replaced by the inverse of the information matrix. In particular, if one tests whether $\theta$ is on one side or the other of a smooth $(k - 1)$-dimensional surface in $k$-dimensional space and $\theta$ lies on the surface, the asymptotic distribution of $\lambda$ is that of a chance variable which is zero half the time and which behaves like $\chi^2$ with one degree of freedom the other half of the time.