Moment generating function based estimators with some optimal properties

In this paper, a behavior of the moment generating function based estimator for the natural parameter θ of a natural exponential family on R is studied. This estimator, say θ̂_n,s, depends on the sample size n and on an auxiliary variable s controlled by the experimenter, and is obtained as a solution of an equation generated by equating the theoretical moment generating function with its empirical counterpart. For fixed n, necessary and sufficient conditions for the existence and uniqueness of θ̂_n,s with probability 1 are presented. Asymptotically it is shown that for any fixed s,θ̂ _n,s is strongly consistent for θ as n→∞; and for any fixed n,θ̂ _n,s converges to the maximum likelihood estimator for θ as s→0. Moreover, under suitable normalization, the limiting distribution of θ̂_n,s, as either n→∞ and s→0, or as s→0 and n→∞, is shown to coincide with that of the maximum likelihood estimator. Such asymptotic results suggest, in some situations, the use of θ̂_n,s for large values of n and small values of s as an alternative to the ordinary maximum likelihood estimator.