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A graded scale of parametric families of distributions, and parameter estimates based on the sample mean

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Summary

Denote by k a class of familiesP={Pθ} of distributions on the line R1 depending on a general scalar parameter θεΘ, Θ being an interval of R1, and such that the moments µ1(θ)=∫xdP θ,...,µ2k (θ)=∫x 2k dP θ are finite, μ1‴ (θ), ..., μk‴ (θ), μk+1 ″ (θ) ..., μ k ″ (θ) exist and are continuous, with μ1′ (θ) ≠ 0, and μ j +1 (θ)= μ1 (θ)μ j (θ) +[μ2(θ) -μ1(θ)2 j ′ (θ)/ μ1′ (θ), J=2, ..., k. Let μ1x=x 1 + ... +x n/n, α2=x 1 2 + ... +x n 2/n, ..., α k =(x 1 k + ... +x n k/n denote the sample moments constructed for a sample x1, ..., xn from a population with distribution Pg. We prove that the estimator of the parameter θ by the method of moments determined from the equation α1= μ1(θ) and depending on the observations x1, ..., xn only via the sample mean ¯x is asymptotically admissible (and optimal) in the class ℐ k of the estimators determined by the estimator equations of the form λ0 (θ) + λ1 (θ) α1 + ... + λ k (θ) α k =0 if and only ifP k .

The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n → ∞ (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every θ).

The scales arise of classes 1 2⊃... of parametric families and of classes ℐ1⊂ ℐ2 ⊂ ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class κ k is equivalent to the membership of the familyP in the class k .

The intersection

consists only of the families of distributions with densities of the form h(x) exp {C0(θ) + C1(θ) x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments μ1 (θ), μ2 (θ), ...

Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.

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Literature cited

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Additional information

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.

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Kagan, A.M. A graded scale of parametric families of distributions, and parameter estimates based on the sample mean. J Math Sci 34, 1482–1487 (1986). https://doi.org/10.1007/BF01089785

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Keywords

  • Parameter Estimate
  • Location Parameter
  • Parametric Family
  • Estimator Equation