Bayesian predictive densities for linear regression models under \(\alpha\)-divergence loss: Some results and open problems. (English) Zbl 1326.62021

Maruyama, Yuzo; Strawderman, William E.

Fourdrinier, Dominique (ed.) et al., Contemporary developments in Bayesian analysis and statistical decision theory. A festschrift for William E. Strawderman. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-81-2). Institute of Mathematical Statistics Collections 8, 42-56 (2012).

Summary: This paper considers estimation of the predictive density for a normal linear model with unknown variance under \(\alpha\)-divergence loss for \(-1\leq\alpha\leq 1\). We first give a general canonical form for the problem, and then give general expressions for the generalized Bayes solution under the above loss for each \(\alpha\). For a particular class of hierarchical generalized priors studied in [the authors, Ann. Stat. 33, No. 4, 1753–1770 (2005; Zbl 1078.62006); J. Stat. Plann. Inference 136, No. 11, 3822–3836 (2006; Zbl 1104.62005)] for the problems of estimating the mean vector and the variance respectively, we give the generalized Bayes predictive density. Additionally, we show that, for a subclass of these priors, the resulting estimator dominates the generalized Bayes estimator with respect to the right invariant prior, i.e., the best (fully) equivariant minimax estimator when \(\alpha=1\).
For the entire collection see [Zbl 1319.62003].

MSC:

62C20	Minimax procedures in statistical decision theory
62J07	Ridge regression; shrinkage estimators (Lasso)
62F15	Bayesian inference

Keywords:

shrinkage prior; Bayesian predictive density; alpha-divergence; Stein effect

Citations:

Zbl 1078.62006; Zbl 1104.62005