Regularity Conditions for Cox’s Test of Non-nested Hypotheses

Often it is desired to test a composite null hypothesis against a composite alternative that is not in the same parametric family as the null hypothesis. For example, suppose we have independent, identically distributed (i.i.d.) observations U,, .., U, and we wish to test the hypothesis that these observations arise from a normal distribution against the alternative that they rise from a logistic distribution. Several procedures are available for dealing with this kind of problem. By far the most commonly used approach is that proposed by Cox (1961, 1962). In these classic papers, Cox proposes that we maximize the likelihood function under both the null and alternative hypotheses, and from these, form a log-likelihood ratio. The value of this log-likelihood ratio is then compared to the value expected when the null hypothesis is true. Small deviations from the expected value are evidence in favor of the null hypothesis, while large deviations are evidence against. Cox provides a statistic which is heuristically argued to be distributed asymptotically as unit normal. However, Cox (1961, p. 105; 1962, p. 408) explicitly eschews attempting to provide regularity conditions which ensure the general validity of his procedure. In spite of the widespread application of Cox’s test, general regularity conditions and a rigorous proof of the asymptotic normality of Cox’s statistic have yet to be given. [Regularity conditions in a special case have been given by Amemiya (1973).] In this article, we rectify this omission. The results are useful since they provide a straightforward way to establish the validity of