THE VARIATIONAL FORM OF CERTAIN BAYES ESTIMATORS
THE VARIATIONAL FORM OF CERTAIN BAYES ESTIMATORS
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DOI：
10.1214/aos/1176348244
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发表时间：
1991-09-01
影响因子：
4.5
通讯作者：
HAFF, LR
中科院分区：
文献类型：
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作者：
HAFF, LR
给出了参数矩阵形式Bayes估计的一般表示。 我们假设先验分布在某种意义上是对称的，但没有另外指定。 形式贝叶斯风险最小化的顺序约束的变分技术，因此我们的表示被称为“变分形式的贝叶斯估计”（VFBE）。 VFBE用于获得相对于通常估计量具有良好频率特性的估计量。 这样的估计得到的均值向量和协方差矩阵的多元正态分布。 此外，对于可能的非正态数据，我们给出了几个皮尔逊均值的VFBE。 一定的重点放在估计的协方差矩阵的问题。 对于这个问题，我们的约束优化提供了一个具有非常好的性质的估计：它的特征值是在适当的顺序，他们不像那些在样本协方差矩阵失真。 协方差矩阵的VFBE与Stein的估计量有关。 在两者中，VFBE以更自然的方式处理顺序关系;也就是说，它更依赖于标准。 此外，它比Stein估计更容易计算，并且简单的Monte Carlo模拟表明它也具有更好的风险特性。
A general representation is obtained for the formal Bayes estimator of a parameter matrix. We assume that the prior distribution is symmetric in some sense, but it is not specified otherwise. The formal Bayes risk is minimized subject to order constraints by a variational technique; hence our representation is called "the variational form of the Bayes estimator" (VFBE). The VFBE is used to obtain estimators that have good frequency properties relative to the usual estimators. Such estimators are obtained for the mean vector and covariance matrix of a multivariate normal distribution. Also, for possibly nonnormal data, we give the VFBE of several Pearson means. A certain emphasis is placed on the problem of estimating the covariance matrix. For that problem, our constrained optimization provides an estimator with very good properties: Its eigenvalues are in the proper order, and they are not as distorted as those in the sample covariance matrix. The VFBE for the covariance matrix is related to an estimator of Stein. Of the two, the VFBE deals with order relations in a more natural way; that is, it is more criterion dependent. In addition, it is easier to compute than Stein's estimator, and a brief Monte Carlo simulation indicates that it has better risk properties as well.