On improved shrinkage estimators for concave loss

On shrinkage estimation for balanced loss functions

“…The proof of Theorem 3.3 is unified with respect to the choice of ρ, the coefficient ω in the balanced loss, and the underlying scale mixture or normals distribution. To conclude, we point out that the above result can be seen as extensions of Kubokawa et al (2015), as well as Strawderman (1974), whose results can be seen as particular cases of ω = 0 in the former case, and ω = 0, ρ(t) = t in the latter case.…”

“…For balanced loss L ω,ρ with ρ satisfying Assumption 1, a scale mixture of normals distribution on X with d ≥ 3, we provide James-Stein and Baranchick-type estimators that dominate X. In such cases, it follows that X is minimax for the unbalanced case L 0,ρ with constant risk R 0 (e.g., Kubokawa et al, 2015). By virtue of Theorem 2.2, X is also minimax for balanced loss L ω,ρ .…”

“…For the unbalanced case ω = 0, one recovers Theorem 2.1 ofKubokawa et al (2015). For the original balanced loss function with ℓ(t) = t, one may recover the result of Theorem 4.4 directly by relying on Lemma 4.7 and Lemma 2.2, as illustrated in Example 2.1 for the multivariate normal case.Moreover, it is interesting to compare the balanced and unbalanced cut-off points a 0…”

“…Shrinkage estimation for multivariate normal models, and more generally spherically symmetric and elliptically symmetric models, has had a long, rich and influential history (e.g., Fourdrinier et al, 2018). The use of a concave loss as well as concave versions of (1.2) and (1.3), is quite appealing, and has motivated previous shrinkage estimation work such as Brandwein & Strawderman (1980, 1991, Brandwein et al (1993), and Kubokawa et al (2015), among others.…”

Shrinkage estimation of non-negative mean vector with unknown covariance under balance loss

“…Marchand and Strawderman [16] developed a unified approach for minimax estimation for a restricted parameter space. Kubokawa et al [13] considered minimax shrinkage estimation of a location vector of a spherically symmetric distribution under a concave squared error loss. Also Chang and Strawderman [3] studied a shrinkage estimation of p positive normal means under sum of squared errors loss.…”