Moment generating function based estimators with some optimal properties

Abstract

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.