Universal Domination and Stochastic Domination: Estimation Simultaneously Under a Broad Class of Loss Functions
Universal Domination and Stochastic Domination: Estimation Simultaneously Under a Broad Class of Loss Functions
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普遍支配和随机支配：在广泛的损失函数类别下同时估计
DOI：
10.1214/aos/1176346594
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发表时间：
1985
期刊：
影响因子：
--
通讯作者：
J. T. Hwang
中科院分区：
文献类型：
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作者：
J. T. Hwang
本文解决了几个理论问题。特别地，证明了泛控制准则等价于随机控制准则，即通过估计量到真实参数的欧氏距离的随机排序来比较估计量。还建立了与常用估计量有关的泛控制的具体结果。特别地，当X-0具有p元t分布，且p=1，2时，不存在普遍控制X的0的估计量；然而，对于p23，指定了普遍控制X的(James-Stein正部估计类型的)估计。当X具有均值为0且具有单位协方差矩阵的p元正态分布时，我们证明了对于任意维p，没有James-Stein正部估计普遍控制X.然而，在较小的损失类别下，证明了某些James-Stein正部估计同时控制X.这些迄今未被研究的损失是有界的且相当实用.
Several theoretical questions are resolved in this paper. In particular the criterion of universal domination is shown to be equivalent to the criterion of stochastic domination that compares the estimators by the stochastic ordering of their Euclidean distances from the estimators to the true parameter. Concrete results about universal domination relating to the usual estimator are also established. In particular when X - 0 has a p-variate t distribution, and p = 1, 2, there exists no estimator for 0 that universally dominates X; however, for p 2 3, estimators (of the type of James-Stein positive part estimators) that universally dominate X are specified. When X has a p-variate normal distribution with mean 0 and identity covariance matrix, we show that for any dimension p, no James-Stein positive part estimators universally dominate X. However, under slightly smaller classes of losses, some JamesStein positive part estimators are shown to simultaneously dominate X. These hitherto unstudied losses are bounded and fairly practical.