A unified and generalized set of shrinkage bounds on minimax Stein estimates

Consider the problem of estimating the mean vector θθ of a random variable XX in RpRp, with a spherically symmetric density f(‖x−θ‖2)f(‖x−θ‖2), under loss ‖δ−θ‖2‖δ−θ‖2. We give an increasing sequence of bounds on the shrinkage constant of Stein-type estimators depending on properties of f(t)f(t) that unify and extend several classical bounds from the literature. The basic way to view the conditions on f(t)f(t) is that the distribution of XX arises as the projection of a spherically symmetric vector (X,U)(X,U) in Rp+kRp+k. A second way is that f(t)f(t) satisfies (−1)jf(j)(t)≥0(−1)jf(j)(t)≥0 for 0≤j≤ℓ0≤j≤ℓ and that (−1)ℓf(ℓ)(t)(−1)ℓf(ℓ)(t) is non-increasing where k=2(ℓ+1)k=2(ℓ+1). The case ℓ=0ℓ=0 (k=2k=2) corresponds to unimodality, while the case ℓ=k=∞ℓ=k=∞ corresponds to complete monotonicity of f(t)f(t) (or equivalently that f(‖x−θ‖2)f(‖x−θ‖2) is a scale mixture of normals). The bounds on the minimax shrinkage constant in this paper agree with the classical bounds in the literature for the case of spherical symmetry, spherical symmetry and unimodality, and scale mixtures of normals. However, they extend these bounds to an increasing sequence (in kk or ℓℓ) of minimax bounds.