Admissibility Results for Generalized Bayes Estimators of Coordinates of a Location Vector

Let X be an n-dimensional random vector with density f(x - θ). It is desired to estimate θ₁, under a strictly convex loss L(δ - θ₁). If F is a generalized Bayes prior density, the admissibility of the corresponding generalized Bayes estimator, δF, is considered. An asymptotic approximation to δF is found. Using this approximation, it is shown that if (i) f has enough moments, (ii) L and F are smooth enough, and (iii) F(θ) K(|θ₁| + ∑ⁿᵢ₌₂ θᵢ²)(3-n)/2, then δF is admissible for estimating θ₁. For example, assume that F(θ) ≡ 1 and that L is squared error loss. Under appropriate conditions it can be shown that δF(x) = x₁, and that δF is the best invariant estimator. If, in addition, f has 7 absolute moments and n 3, it can be concluded that δF is admissible.