Moment generating function based estimators with some optimal properties

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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 θ̂n,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 θ̂n,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 θ̂n,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 θ̂n,s for large values of n and small values of s as an alternative to the ordinary maximum likelihood estimator.

Original languageEnglish
Pages (from-to)279-291
Number of pages13
JournalJournal of Statistical Planning and Inference
Issue number3
StatePublished - Jun 1993


  • Moment generating function based estimator
  • asymptotic efficiency
  • maximum likelihood estimator
  • natural exponential family

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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