A Natural Identity for Exponential Families with Applications in Multiparameter Estimation

A random variable X is said to have distribution in the class E₀ if, for some real valued, positive function a(•), the identity E\(X - μ)g(X)\ = E(X)g'(X)\ holds for any absolutely continuous real valued function g(•) satisfying E|a(X)g'(X)| < ∞. Examples of a random variable X possessing a distribution in E₀ are (i) X normally distributed with mean μ and standard deviation 1, (ii) X having a gamma density with mean μ and location parameter 1, (iii) $X = 1/Y$ where Y ~ (n - 2)⁻¹χₙ², n > 2. Suppose X₁,⋯, Xₚ, p 3, are independently distributed with distributions in E₀, for some function a(•), and with means μ₁,⋯, μₚ. Define b(x) = ∫ a(x)⁻¹ dx, where the integral is interpreted as indefinite, Bᵢ = b(Xᵢ), S = ∑ᵖᵢ₌₁ Bᵢ², X' = (X₁,⋯, Xₚ) and B' = (B₁,⋯, Bₚ). Then the estimator $X - ((p - 2)/S)B$ dominates X if sum of squared error loss is assumed. Similar estimators are obtained, when p 4, which shrink towards an origin determined by the data. There are corresponding results for some discrete exponential families.