An extended class of minimax generalized Bayes estimators of regression coefficients

Abstract

a b s t r a c t We derive minimax generalized Bayes estimators of regression coefficients in the general linear model with spherically symmetric errors under invariant quadratic loss for the case of unknown scale. The class of estimators generalizes the class considered in Maruyama and Strawderman [Y. Maruyama, W.E. Strawderman, A new class of generalized Bayes minimax ridge regression estimators, Ann. Statist., 33 (2005) 1753-1770] to include non-monotone shrinkage functions.

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

“…Proof of Theorems 3.3 and 3.4. By following Maruyama and Strawderman (2005) and Maruyama and Strawderman (2009), the generalized Bayes estimator under the prior given by (3.10) is δ φ with…”

“…Strawderman (1973) and Zinodiny, Strawderman and Parsian (2011) gave a class of proper Bayes minimax and hence admissible estimators under unknown σ 2 . Note that proper Bayes estimators by Strawderman (1973) and Zinodiny, Strawderman and Parsian (2011) are not of the form given by (1.4) whereas generalized Bayes estimators by Maruyama (2003), Maruyama and Strawderman (2005) and Maruyama and Strawderman (2009) are of this form.…”

“…The generalized Bayes estimator is quasi-inadmissible.Proof of Theorems 3.3 and 3.4. By followingMaruyama and Strawderman (2005) andMaruyama and Strawderman (2009), the generalized Bayes estimator under the prior given by (3.10) is δ φ withφ a,L (w) = w 1 0 λ p/2+a+1 L(1/λ)(1 + wλ) −(p+n)/2−1 dλ 1 0 λ p/2+a L(1/λ)(1 + wλ) −(p+n)/2−1 dλ . By a change of variables (t = wλ), we have φ a,L (w) = w 0 t p/2+a+1 L(w/t)(1 + t) −(p+n)/2−1 dt w 0 λ p/2+a L(w/t)(1 + t) −(p+n)/2−1 dt .…”

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

“…This estimator is minimax when Maruyama (2003). For Part 3, see Maruyama and Strawderman (2009) and Appendix J.…”

Akaike's Information Criterion, Cp and Estimators of Loss for Elliptically Symmetric Distributions

“…is the canonical form of the linear regression model (1). This canonical form has been considered by various authors such as Cellier, Fourdrinier, and Robert (1989); Cellier and Fourdrinier (1995); Maruyama (2003); Maruyama and Strawderman (2009); Fourdrinier and Strawderman (2010); Kubokawa and Srivastava (1999). Kubokawa and Srivastava (2001) addressed the multivariate case where θ is a mean matrix (in this case Z and U are matrices as well) we will introduce section 4.2.…”