The likelihood ratio under noncontiguous alternatives

The asymptotic distribution of the likelihood ratio under noncontiguous alternatives is shown to be normal for the exponential family of distributions. The rate of convergence of the parameters to the hypothetical value is specified where the asymptotic noncentral chi-square distribution no longer holds. It is only a little slower than $Oleft( {n^{ - frac{1}{2}} } right)$. The result provides compact power approximation formulae and is shown to work reasonably well even for moderate sample sizes.