A rate of convergence for the set compound estimation in a family of certain retracted distributions

Summary This paper is concerned with the set compound squared-error loss estimation problem. Here, the author obtains Lévy consistent estimate of the empiric distributionG _n of the parameters θ₁,...,θ_n for a more general family of retracted distributions on the interval [θ, θ+1) than the uniform on [θ, θ+1) as in R. Fox (1970,Ann. Math. Statist.,41, 1845–1852; 1978,Ann. Statist.,6, 846–853) and exhibits a decision procedure based on with a convergence rateO((n ⁻¹ logn)^1/4) for the mofified regret uniformly in (θ₁, θ₂, ..., θ_n ∈ Ωⁿ with bounded Ω. The author also gives a counterexample to the convergence of the modified regret for Ω=(−∞, ∞). This is part of the author's Ph. D. Thesis at Michigan State University.