On Improved Loss Estimation for Shrinkage Estimators
On Improved Loss Estimation for Shrinkage Estimators
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DOI：
10.1214/11-sts380
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发表时间：
2012-02
影响因子：
5.7
通讯作者：
D. Fourdrinier;M. Wells
中科院分区：
文献类型：
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作者：
D. Fourdrinier;M. Wells
令 X 为具有分布 Pθ 的随机向量，其中 θ 是未知参数。当在损失函数 L(θ, φ) 下通过某个估计器 φ(X) 估计 θ 时，经典决策理论主张，如果这种决策规则对于频率风险 R(θ, φ) 具有合适的属性，则应使用该决策规则。然而，在观察到 X = x 后，实践中会出现 φ 伴随着对其损失 L(θ, φ(x)) 的评估的情况，这是不可观察的，因为 θ 是未知的。此评估的常见方法是考虑通过估计器 δ（称为损失估计器）估计 L(θ, φ(x))。我们提出了损失估计的说明性发展，重点强调分布环境为正态的设置及其扩展到基础分布为球对称的情况。我们的概述涵盖了改进的最小二乘损失估计器，但主要关注收缩估计器。还考虑了贝叶斯估计，并与无偏估计进行比较。
Let X be a random vector with distribution Pθ where θ is an unknown parameter. When estimating θ by some estimator φ(X) under a loss function L(θ, φ), classical decision theory advocates that such a decision rule should be used if it has suitable properties with respect to the frequentist risk R(θ, φ). However, after having observed X = x, instances arise in practice in which φ is to be accompanied by an assessment of its loss, L(θ, φ(x)), which is unobservable since θ is unknown. A common approach to this assessment is to consider estimation of L(θ, φ(x)) by an estimator δ, called a loss estimator. We present an expository development of loss estimation with substantial emphasis on the setting where the distributional context is normal and its extension to the case where the underlying distribution is spherically symmetric. Our overview covers improved loss estimators for least squares but primarily focuses on shrinkage estimators. Bayes estimation is also considered and comparisons are made with unbiased estimation.