Group-Linear Empirical Bayes Estimates for a Heteroscedastic Normal Mean
Group-Linear Empirical Bayes Estimates for a Heteroscedastic Normal Mean
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DOI：
10.1080/01621459.2017.1280406
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发表时间：
2015-03
影响因子：
3.7
通讯作者：
A. Weinstein;Zhuang Ma;L. Brown;Cun-Hui Zhang
中科院分区：
文献类型：
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作者：
A. Weinstein;Zhuang Ma;L. Brown;Cun-Hui Zhang
摘要估计方差已知但不相等的正态向量的均值问题带来了相当大的困难，削弱了传统经验贝叶斯估计的充分性。通过采取一种不同的方法，将已知方差作为随机观测的一部分，我们恢复了对称性，从而恢复了这种方法的有效性。我们提出了一种群线性经验贝叶斯估计，它收集具有相似方差的观测值，并分别对每个组应用球对称估计。所提出的估计器是由一个新的Oracle规则驱动的，该规则比最佳线性规则更强，因此提供了一个比以前文献中考虑的更雄心勃勃的基准。我们的估计量(在适当的条件下)渐近达到新的先知风险，同时也是极小极大估计。在真实均值和观察到的方差是经验依赖的情况下，分组线性估计特别有利。为了证明所提方法在实际应用中的优点，我们分析了Brown(2008)使用的棒球数据，其中群线性方法获得了已应用于数据集的最佳非参数估计的预测误差，并且显著低于其他参数和半参数贝叶斯估计。
ABSTRACT The problem of estimating the mean of a normal vector with known but unequal variances introduces substantial difficulties that impair the adequacy of traditional empirical Bayes estimators. By taking a different approach that treats the known variances as part of the random observations, we restore symmetry and thus the effectiveness of such methods. We suggest a group-linear empirical Bayes estimator, which collects observations with similar variances and applies a spherically symmetric estimator to each group separately. The proposed estimator is motivated by a new oracle rule which is stronger than the best linear rule, and thus provides a more ambitious benchmark than that considered in the previous literature. Our estimator asymptotically achieves the new oracle risk (under appropriate conditions) and at the same time is minimax. The group-linear estimator is particularly advantageous in situations where the true means and observed variances are empirically dependent. To demonstrate the merits of the proposed methods in real applications, we analyze the baseball data used by Brown (2008), where the group-linear methods achieved the prediction error of the best nonparametric estimates that have been applied to the dataset, and significantly lower error than other parametric and semiparametric empirical Bayes estimators.