A sharp boundary for SURE-based admissibility for the normal means problem under unknown scale

Abstract

We consider quasi-admissibility/inadmissibility of Stein-type shrinkage estimators of the mean of a multivariate normal distribution with covariance matrix an unknown multiple of the identity. Quasi-admissibility/inadmissibility is defined in terms of non-existence/existence of a solution to a differential inequality based on Stein's unbiased risk estimate (SURE). We find a sharp boundary between quasi-admissible and quasi-inadmissible estimators related to the optimal James-Stein estimator. We also find a class of priors related to the Strawderman class in the known variance case where the boundary between quasi-admissibility and quasi-inadmissibility corresponds to the boundary between admissibility and inadmissibility in the known variance case. Additionally, we also briefly consider generalization to the case of general spherically symmetric distributions with a residual vector.

Admissible estimators of a multivariate normal mean vector when the scale is unknown

“…The key results under η −1 η p/2 π(η θ 2 | a, b) are summarized as follows and in Figure 1. The estimator is inadmissible if −p/2 − 1 < a < −2 and b > −1 (Maruyama & Strawderman, 2017) and is admissible within the class of scale equivariant estimators if a ≥ −2 and b > −1/2 (Maruyama & Strawderman, 2020). Furthermore the estimator is minimax if −p/2 − 1 < a ≤ ξ(p, n) and b = 0 (Lin & Tsai, 1973) and if −p/2 − 1 < a ≤ ξ(p, n) and b > 0 (Maruyama & Strawderman, 2005), where…”

Admissible Bayes equivariant estimation of location vectors for spherically symmetric distributions with unknown scale

“…where in the first expression {log λ} 1− satisfies Assumption A.3, and in the second, log λ does not satisfy Assumption A.3. Actually, in the third case, {log λ} 1+ , the corresponding Bayes equivariant estimator is inadmissible as shown in Maruyama and Strawderman (2017).…”

On admissible estimation of a mean vector when the scale is unknown

“…The authors, in Maruyama and Strawderman (2017), have earlier shown that such conditional priors with a < −2 lead to inadmissible estimators. This paper therefore completes the determination of admissibility/inadmissibility for this class of priors.…”