Universal Domination and Stochastic Domination: U -Admissibility and U -Inadmissibility of the Least Squares Estimator
Universal Domination and Stochastic Domination: U -Admissibility and U -Inadmissibility of the Least Squares Estimator
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普遍支配和随机支配：最小二乘估计量的 U 可接受性和 U 不可接受性
DOI：
10.1214/aos/1176347014
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发表时间：
1989
期刊：
影响因子：
--
通讯作者：
J. T. Hwang
中科院分区：
文献类型：
--
作者：
L. Brown;J. T. Hwang
假设标准线性模型Xn×1 = An×p θp×1 + εn×1，其中ε服从正态分布，均值向量为零，协方差矩阵为单位。系数θ的最小二乘估计为θ^<$（A′A）−1A′X。众所周知，在平方误差损失和下，θ^被James-Stein型估计所控制|θ−θ^|当p ≥ 3时为2。在这篇文章中，我们讨论了改进θ^的可能性，同时在“通用”损失类别下：{L（|联系我们|）：L（.）任何非减函数}一个可以如此改进的估计量被称为普遍不可接受（U-不可接受）。否则，它被称为U-可受理。证明了当A′A = I时，θ^对任意p都是U-容许的.此外，如果A′A <$I，则θ^是U-不可容许的，如果p“足够大”。“在特殊情况下，p ≥ 4足够大。结果令人惊讶。的影响进行了讨论。
Assume the standard linear model Xn×1 = An×p θp×1 + εn×1, where ε has an n-variate normal distribution with zero mean vector and identity covariance matrix. The least squares estimator for the coefficient θ is θ^ ≡ (A′A)−1A′X. It is well known that θ^ is dominated by James-Stein type estimators under the sum of squared error loss |θ−θ^|2 when p ≥ 3. In this article we discuss the possibility of improving upon θ^, simultaneously under the "universal" class of losses: {L(|θ θ^|) : L (.) any nondecreasing function} An estimator that can be so improved is called universally inadmissible (U-inadmissible). Otherwise it is called U-admissible. We prove that θ^ is U-admissible for any p when A′A = I. Furthermore, if A′A ≠ I, then θ^ is U-inadmissible if p is "large enough." In a special case, p ≥ 4 is large enough. The results are surprising. Implications are discussed.