On the asymptotic distribution of the likelihood ratio under the regularity conditions due to doob

Summary The concepts of the asymptotic maximum likelihood estimates—AMLEs in short—and their asymptotic identity are introduced in section 1. They seem to be more adequate than the usual one for uses in the large sample theory. The AMLE is a slightly weakened version of the usual maximum likelihood estimate and therefore it should have a bit wider applicability than the original one. The asymptotic normality of a consistent AMLE and Wilks’ theorem concerning the asymptotic distribution of the statistic —2 log λ, where λ is the likelihood ratio, can be obtained under the regularity conditions due to Doob in section 2. A set of conditions which assure the existence of a unique and consistent AMLE is presented in section 3 and in the final section 4 the proof of the existence of the unique and consistent AMLE under those conditions is given. This work has been motivated by the work of Ogawa, Moustafa and Roy 3].