Is there an example of a parameterized model (i.i.d.) in which the maximum likelihood estimate is consistent (e.g., in probability) but not asymptotically efficient, but there exists a sequence of stationary points of the likelihood function that is asymptotically efficient? Alternatively, does anyone know of a theorem that precludes this possibility? For example, Theorem 5.1 of Lehmann & Casella, 2nd ed., p. 463 does not rule out such a possibility even when the usual regularity conditions are satisfied. In particular, that theorem does not say anything about the maximum likelihood estimate itself, only stationary points of the likelihood function. This would seem to be a rather fundamental question in classical statistics, and if such behavior were not possible, existing standard theorems (such as Theorem 5.1 above) would say something stronger than they do.
Thanks!