Estimation, prediction and the Stein phenomenon under divergence loss
Estimation, prediction and the Stein phenomenon under divergence loss
复制标题
散度损失下的估计、预测和斯坦因现象
DOI：
10.1016/j.jmva.2008.02.002
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发表时间：
2008
影响因子：
1.6
通讯作者：
G. Datta
中科院分区：
文献类型：
--
作者：
M. Ghosh;V. Mergel;G. Datta
我们考虑两个问题：（1）在[S]中引入的一般发散损失下，估计正态平均。曲指数族的微分几何-曲率和信息损失，统计学家。10（1982）357-387]和[N. Cressie，T.R.C. Read，Multinomial goodness-of-fit tests，J. Roy.中央集权主义者Soc. Ser. B. 46（1984）440-464]和（2）找到独立于从具有相同均值但在相同损失下可能具有不同方差的正态分布中采样的观测而绘制的新观测的预测密度。一般的发散损失包括Kullback-Leibler和Bhattacharyya-Hellinger损失。样本平均值，这是一个贝叶斯估计的人口平均在这种损失和不适当的统一前，被证明是极小极大在任何任意尺寸。对应的预测密度的结果也证明了在任何任意尺寸。这些规则的可接受性在一维中成立，我们推测，结果是真实的二维。然而，一般Baranchick [A. J. Baranchick，多元正态分布均值的极大极小估计族，Ann. Math. Statist. 41（1970）642-645]类估计量，其包括James-Stein估计量和Strawderman [W.E.张文，张文，张文，等.多元正态均值的贝叶斯最小最大估计.统计学报，2000，24（1）：117 - 118. 42（1971）385-388]类估计量，在估计问题的三个或更高维度上支配样本均值。一个类似的类的预测密度的定义和这一类的任何成员示出占主导地位的预测密度对应于一个统一的前在三个或更高的维度。对于预测问题，在Kullback-Leibler损失的特殊情况下，我们的结果在一定程度上补充了Komaki [F. Komaki，A shrinkage predictive distribution for multivariate normal observations，Biometrika 88（2001）859-864]和乔治，Liang和Xu [E.I.乔治，F. Liang，X. Xu，Kullbak-Leibler损失下改进的极大极小预测密度，Ann. Statistist。34（2006）78-92]。虽然我们提出的方法产生了一个一般类的预测密度（不一定是贝叶斯，但不排除贝叶斯预测）主导的预测密度下一个统一的先验。我们还表明，各种修改的James-Stein估计继续占主导地位的样本平均值，并通过估计和预测密度的结果，我们将显示的对偶，类似的结果继续持有的预测问题，以及。
We consider two problems: (1) estimate a normal mean under a general divergence loss introduced in [S. Amari, Differential geometry of curved exponential families — curvatures and information loss, Ann. Statist. 10 (1982) 357–387] and [N. Cressie, T.R.C. Read, Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. Ser. B. 46 (1984) 440–464] and (2) find a predictive density of a new observation drawn independently of observations sampled from a normal distribution with the same mean but possibly with a different variance under the same loss. The general divergence loss includes as special cases both the Kullback–Leibler and Bhattacharyya–Hellinger losses. The sample mean, which is a Bayes estimator of the population mean under this loss and the improper uniform prior, is shown to be minimax in any arbitrary dimension. A counterpart of this result for predictive density is also proved in any arbitrary dimension. The admissibility of these rules holds in one dimension, and we conjecture that the result is true in two dimensions as well. However, the general Baranchick [A.J. Baranchick, a family of minimax estimators of the mean of a multivariate normal distribution, Ann. Math. Statist. 41 (1970) 642–645] class of estimators, which includes the James–Stein estimator and the Strawderman [W.E. Strawderman, Proper Bayes minimax estimators of the multivariate normal mean, Ann. Math. Statist. 42 (1971) 385–388] class of estimators, dominates the sample mean in three or higher dimensions for the estimation problem. An analogous class of predictive densities is defined and any member of this class is shown to dominate the predictive density corresponding to a uniform prior in three or higher dimensions. For the prediction problem, in the special case of Kullback–Leibler loss, our results complement to a certain extent some of the recent important work of Komaki [F. Komaki, A shrinkage predictive distribution for multivariate normal observations, Biometrika 88 (2001) 859–864] and George, Liang and Xu [E.I. George, F. Liang, X. Xu, Improved minimax predictive densities under Kullbak–Leibler loss, Ann. Statist. 34 (2006) 78–92]. While our proposed approach produces a general class of predictive densities (not necessarily Bayes, but not excluding Bayes predictors) dominating the predictive density under a uniform prior. We show also that various modifications of the James–Stein estimator continue to dominate the sample mean, and by the duality of estimation and predictive density results which we will show, similar results continue to hold for the prediction problem as well.