Admissibility of an estimate obtained by the method of moments in the presence of a nuisance shift parameter

Abstract

Let x₁,..., x_n be a repeated sample from a one-dimensional population with distribution function (d.f.) F(x−η, θ), depending on a structure parameter θ∈^Θ⊂R ¹ and a nuisance shift parameter η^∈ R¹. The estimator which eliminates ν In a natural manner, has the form$\sum\limits_1^n {\psi (x_i - \overline x ,\theta ) = 0,\overline x = (x_1 + ... + x_n )/n}$ and the simplest among them, corresponding to a functionψ (u, θ), quadratic in u, leads to the estimate θ (m₂), where$m_2 = \sum\limits_1^n {(x_i - \overline x )^2 /n}$ which has to be considered as an estimate of θ by the method of moments with the elimination of the nuisance parameter n. If for some integer k ≥ 1, 1°) the d.f. F(x, θ) has a finite moment of order 2k, 2°) its central moments μ₂(θ), ..., μk(θ) are three times and μk+1(9).... μ₂k(θ) are twice continuously differentiable in the domain Θ and μ₂′(θ) ≠ 0, 3° as n → ∞, the limit covariance matrix of the centralized and normalized vector √n ∥ m₂-μ₂(θ) ...,m_R-μ_R(θ)∥ of the central sample moments m_j is nonsingular, θ∈^Θ, then the estimate θ(m₂) is asymptotically admissible (and optimal) in the class of estimates defined by the estimators λ_o(θ) + λ₂(θ)m₂ + ... + λ_k(θ)m_k=0 if and only if the moments μ₅(θ),..., μk+2 (θ) are determined in terms of μ₂(θ), μ₃(θ), μ₄(θ) in the following recurrent manner;

$$\begin{array}{*{20}c} {\mu _{j + 2} (\theta ) = \mu _2 (\theta )\mu _j (\theta ) + j\mu _3 (\theta )\mu _{j - 1} (\theta ) + [\mu _4 (\theta ) - \mu _2 (\theta )^2 ]\mu _j ^\prime (\theta )/\mu _2 ^\prime (\theta ),} \\ {j \leqslant k,\theta ^\Theta .} \\ \end{array}$$

The asymptotic admissibility is understood in the same generally accepted sense as in [1], where a similar result has been obtained for families of d.f. containing only a structure parameter.