过度乐观：由SURE调整的估计值的表观误差有多大偏差？
Excess Optimism: How Biased is the Apparent Error of an Estimator Tuned by SURE?
摘要：
几乎所有统计预测中的估计器都以某种方式与相关的调整参数相关联。通常的做法，给定的数据，是选择调整参数值，最大限度地减少估计量的预测误差的构造估计，我们专注于斯坦的无偏风险估计，或SURE（斯坦，1981年;埃夫隆，1986年），形成一个无偏估计的预测误差，通过增加观察到的训练误差估计的自由度。通过SURE最小化的参数调整已经提倡许多作者，在各种各样的问题设置，在一般情况下，这是很自然的问题：什么是SURE调整估计的预测误差？一个明显的策略是简单地使用SURE报告的表观误差估计，即，SURE准则在其最小值处的值，以估计SURE调谐估计器的预测误差。但这不再是无偏的;事实上，我们可以预期SURE准则的最小值对于真实的预测误差是系统性地向下偏置的。在本文中，我们正式描述和研究这种偏见。
英文摘要：
Nearly all estimators in statistical prediction come with an associated tuning parameter, in one way or another. Common practice, given data, is to choose the tuning parameter value that minimizes a constructed estimate of the prediction error of the estimator; we focus on Stein's unbiased risk estimator, or SURE (Stein, 1981; Efron, 1986) which forms an unbiased estimate of the prediction error by augmenting the observed training error with an estimate of the degrees of freedom of the estimator. Parameter tuning via SURE minimization has been advocated by many authors, in a wide variety of problem settings, and in general, it is natural to ask: what is the prediction error of the SURE-tuned estimator? An obvious strategy would be simply use the apparent error estimate as reported by SURE, i.e., the value of the SURE criterion at its minimum, to estimate the prediction error of the SURE-tuned estimator. But this is no longer unbiased; in fact, we would expect that the minimum of the SURE criterion is systematically biased downwards for the true prediction error. In this paper, we formally describe and study this bias.
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