Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

Estimation of location parameters for spherically symmetric distributions

“…Brown [7] proved that the best invariant estimator of a location vector is inadmissible for a wide class of distributions and loss functions if the dimension is at least three. James and Stein's [13] result remains true if the distribution of X is spherically symmetric and p 4 as shown by Brandwein [2], Brandwein and Strawderman [3,4,6], Fan and Fang [11] and others; see the review article by Brandwein and Strawderman [5]. Under the assumption that the components of X are independent, identically and symmetrically (iis) distributed about their respective means, Shinozaki [15] investigated the bounds of a and b in (1) which involve the second and the fourth moments of the component distributions.…”

“…Suppose that X ∼ SS p ( , I p ) (spherically symmetric about ) and a (X) is defined by (2). Then with respect to the quadratic loss (3), a (X) has smaller risk than…”

“…Their idea to show dominance of a (X) over X is to use the fact that if −h is superharmonic, its average over the ball ("volume") is greater than its average over the sphere ("surface area"). The estimators a (X) given by (2), together with conditions (i) and (iii) extend the classical James-Stein estimator to a broader class of estimators, while their condition (ii) is a technical condition. To our knowledge, their result is to date the most general for spherically symmetric distributions.…”

Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family

“…Therefore the generalized Bayes estimator is minimax if p for p 3 by Berger's [2] conditions and if 4 − p for p 4 by Brandwein's [7] conditions regardless of . Hence the estimator for p 4 is minimax regardless of and .…”

“…Berger [2] p 3 2 (p − 2) inf s∈U F (s)/f (s) Brandwein [7] p 4 2(p − 2)(pE 0 ( X −2 )) −1 Unimodal or f is nonincreasing Brandwein and Strawderman [8] p 4 2p((p + 2)E 0 ( X −2 )) −1 Ralescu et al [21] p = 3 0.93(E 0 ( X −2 )) −1 F (t)/f (t) is nondecreasing Bock [6] p 4 2(E 0 ( X −2 )) −1 Scale mixtures of multivariate normal Strawderman [24] p 3 2(E 0 ( X −2 )) −1 which is nonnegative for any f. The derivative of * (r)/r 2 is calculated as…”