On the Distribution of the Likelihood Ratio

In an investigation of the distribution of the likelihood ratio λ, Wilks [3] proved, under certain regularity conditions, that -2 ln λ is, except for terms of order 1/√ n, distributed like χ² with $k - m$ degrees of freedom, where k is the dimension of the parameter space Ω of admissible hypotheses and m is the dimension of the parameter space ω 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 λ has an exact χ²-distribution with $2(k - m)$ degrees of freedom.