From Fixed-X to Random-X Regression: Bias-Variance Decompositions, Covariance Penalties, and Prediction Error Estimation
从固定设计到随机设计回归：偏差方差分解、协方差惩罚与预测误差估计AI 解读 ↓
← 返回论文列表与筛选
阅读说明
本页整理公开书目信息，不生成学术质量评分。AI 解读根据题名及所注明的摘要或原文片段生成，未完成全文核验，可能有误；具体假设、证明与结论请核对原文。摘要保留来源文本，未经模型改写。
DOI 首次登记不等于接收或发表日期。部分更正、评论等条目没有独立摘要；缺少可靠材料时保留待补状态。
引用量采用 OpenAlex 单条关联记录，不合计多个记录；关联记录可能包含正式发表版本，不等于此版本单独引用。缺失不视为零，引用指标不代表论文质量。
关注作者按书目中的完整姓名匹配，同名和姓名变体需人工辨别。相关论文先按共享细分方向和具体方法排列，再参考通用方法与发表日期。
分类依据与纠错
AI 根据所提供的论文材料归类：摘要研究一般回归预测误差与模型评价，没有指向宏观预测或金融收益预测的明确依据。
反馈分类问题
正在核对关联版本…
代码、数据与补充材料
正在核对对应版本的资源…
关注这篇论文的作者
从固定设计到随机设计回归：偏差方差分解、协方差惩罚与预测误差估计
- 研究问题
- 当训练与预测的协变量均随机抽取时，预测误差如何区别于固定设计情形，传统协方差惩罚应怎样调整？
- 方法与思路
- 作者分解随机设计预测误差，重点分析 OLS，并构造 RCp 及其变体，同时考察岭回归的随机设计误差。
- 材料陈述的贡献
- 摘要称在所研究的 OLS 设置下，随机设计增加偏差与方差；提出误差估计准则，并发现强正则化岭回归的不同现象。
- 阅读关注点
- 核对训练与预测抽样设定、各准则的分布和模型条件，以及 OLS 与岭回归结论为何依赖不同的正则化情形。
- 阅读边界
- 依据版本未注明的 OpenAlex 索引摘要；部分适用分布与渐近条件未展开，数值优势仅为摘要陈述，不能推断刊本变化。
原文摘要
In statistical prediction, classical approaches for model selection and model evaluation based on covariance penalties are still widely used. Most of the literature on this topic is based on what we call the “Fixed-X” assumption, where covariate values are assumed to be nonrandom. By contrast, it is often more reasonable to take a “Random-X” view, where the covariate values are independently drawn for both training and prediction. To study the applicability of covariance penalties in this setting, we propose a decomposition of Random-X prediction error in which the randomness in the covariates contributes to both the bias and variance components. This decomposition is general, but we concentrate on the fundamental case of ordinary least-squares (OLS) regression. We prove that in this setting the move from Fixed-X to Random-X prediction results in an increase in both bias and variance. When the covariates are normally distributed and the linear model is unbiased, all terms in this decomposition are explicitly computable, which yields an extension of Mallows’ Cp that we call RCp. RCp also holds asymptotically for certain classes of nonnormal covariates. When the noise variance is unknown, plugging in the usual unbiased estimate leads to an approach that we call RCp ^, which is closely related to Sp, and generalized cross-validation (GCV). For excess bias, we propose an estimate based on the “shortcut-formula” for ordinary cross-validation (OCV), resulting in an approach we call RCp+. Theoretical arguments and numerical simulations suggest that RCp+ is typically superior to OCV, though the difference is small. We further examine the Random-X error of other popular estimators. The surprising result we get for ridge regression is that, in the heavily regularized regime, Random-X variance is smaller than Fixed-X variance, which can lead to smaller overall Random-X error. Supplementary materials for this article are available online.