On Some Bayesian Solutions of the Neyman-Scott Problem
On Some Bayesian Solutions of the Neyman-Scott Problem
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关于内曼-斯科特问题的一些贝叶斯解决方案
DOI：
10.1007/978-1-4612-2618-5_20
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发表时间：
1994
影响因子：
4.5
通讯作者：
M. Ghosh
中科院分区：
文献类型：
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作者：
M. Ghosh
Neyman和Scott（1948）的两个著名例子之一是，在具有正态同方差的固定效应单因素方差分析模型中，当单元格大小保持不变，但单元格数增长到无穷大时，误差方差的极大似然估计σ 2是不一致的。UMVUE或误差平方和的最佳多重估计量不会受到这个缺点的影响。然而，最佳多重估计量是不可接受的，因为它由斯坦型估计量主导。另一方面，Stein估计量是非光滑的，它本身是不可接受的。本文引入了σ 2的一类分层Bayes（HB）估计，并在相对平方误差损失L（a，σ 2）=（a σ-2-1 <$2）下，证明了其中每个成员均优于S（S为误差平方和）的最佳倍数的一个子类.我们的类中包括σ 2的Brewster-Zidek（1974）估计。此外，我们还提供了HB估计的风险改善的最佳多重估计的解析表达式。在某些特殊情况下，给出了风险改善百分比的数值。这些计算表明，最佳多重估计的风险改善往往是相当可观的。
One of the two celebrated examples of Neyman and Scott (1948) is that in a fixed effects one-way analysis of variance model with normal homoscedastic errors, the maximum likelihood estimator of the error variance, say σ 2 is inconsistent as the cell-size remains fixed, but the number of cells grows to infinity. The UMVUE, or the best multiple estimator of the error sum of squares does not suffer from this drawback. However, the best multiple estimator is inadmissible, as it is dominated by a Stein-type estimator. The Stein-estimator, on the other hand, being non-smooth, is itself inadmissible. The present paper introduces a class of hierarchical Bayes (HB) estimators of σ 2, and identifies a subclass, each member of which dominates the best multiple of S (S being the error sum of squares) under the relative squared error loss L(a, σ 2) = (a σ -2 -1)2. Included in our class is the Brewster-Zidek (1974) estimator of σ 2. Also, we have provided analytic expressions for the risk improvement of the HB estimators over the best multiple estimator. Numerical values of the percentage risk improvement are given in some special cases. These calculations indicate that the risk-improvement over the best multiple estimator can often be quite substantial.