On SURE-Type Double Shrinkage Estimation

Abstract

The paper is concerned with empirical Bayes shrinkage estimators for the heteroscedastic hierarchical normal model based on Stein's unbiased estimate of risk (SURE).Recently, Xie, Kou and Brown (2012) proposed a class of estimators for this type of problems and established their asymptotic optimality properties under the assumption of known but unequal variances. In this paper, we consider this problem with unequal and unknown variances, which may be more appropriate in real situations. Specifically, by putting priors for both means and variances, we propose novel double-shrinkage SURE type estimators that shrink both the means and the variances. Optimal properties for these estimators are derived under certain regularity conditions. We also conduct extensive simulation studies to compare the newly developed methodologies with other shrinkage techniques. Finally, the methods are applied to the well known baseball data set and a gene expression data set.

Estimating the mean and variance of a high-dimensional normal distribution using a mixture prior

“…Figure 2 shows that our estimate outperforms the other three when estimating µ j . Table 2 shows that N Γ −1 performs similarly to the double shrinkage algorithms discussed in Jing et al (2016). As the latter algorithms and N Γ −1 are based on the normal-inverse gamma distribution, and the (µ j , σ 2 j ) distribution in this case is normal-inverse gamma, it is not surprising that these estimates outperform the others here.…”

SURE estimates for high dimensional classification

“…With the new samples (Z j , S j ), j = 1, … , p, by using the double shrinkage SURE methods, we have the double SURE-type shrinkage estimators (12) that shrink towards the grand mean (14) and data driven place aŝ…”

“…Next, we present the proposed classification rules based on SURE shrinkage estimates as follows. 18, or the semiparametric SURE-type estimators in Jing et al [14], without additional difficulty. When the variances of X j s' are known, we can apply the SURE estimates of Xie et al [22] to model (7), then the SURE estimates of Equation (9) can be applied and we obtain…”

“…In many real problems, besides the means, the variances in the hierarchical model (7) are also unknown. For this case, Jing et al [14] revisited the SURE estimate of Xie et a. (2012), and proposed the so-called double shrinkage SUER estimators that shrink both means and variances.…”

“…The basic idea of SURE method is to estimate the hyperparameters via minimizing Stein's unbiased estimate of the mean squared risk, then plug these estimators into the Bayes estimators of the unknown parameters. Subsequently, Jing et al [14] studied the same problem as in Xie et al [22], and extended the hierarchical model with known variances to the case of unknown variances. Furthermore, the authors introduced new double shrinkage SURE-type estimators that shrink both means and variances.…”