On the conditions for consistency and asymptotic efficiency of maximum likelihood estimates

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Abstract
Extract Cramér [1] has given sufficient conditions for consistency and asymptotic efficiency of the root of the likelihood equation (Theorem 1 below). In this paper we give two new theorems concerning the same asymptotic properties. Theorem 2 is a direct generalization of Cramér's theorem, in that one of his conditions is replaced by a weaker one. Theorem 3 contains an alternative set of conditions: in contrast to Theorems 1 and 2 it does not assume that the third partial derivative of the density function with respect to the parameter exists, and the consistency property does not even require the existence of the second partial derivative. Also considered is an example which does not satisfy one of Cramér's conditions but which satisfies both the weaker set of conditions in the generalized Theorem 2 and the new set of conditions in Theorem 3.

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