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In an investigation of the distribution of the likelihood ratio, Wilks 3 proved, under certain regularity conditions, that -2 is, except for terms of order 1/ n, distributed like ² with k - m degrees of freedom, where k is the dimension of the parameter space of admissible hypotheses and m is the dimension of the parameter space of null hypotheses. In this paper, we consider the nonregular densities investigated by R. C. Davis 1 and show that for certain hypotheses -2 has an exact ²-distribution with 2 (k - m) degrees of freedom.
Robert V. Hogg (Fri,) studied this question.