发表机构
Pediatric Research Institute; Guangzhou Women and Children’s Medical Center; Guangzhou Medical University（儿科研究所; 广州市妇女儿童医疗中心; 广州医科大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将等变准则从固定设计矩阵扩展到随机协变量情形，区分条件与绝对评估，证明在绝对评估下不存在一致最小风险等变估计量，并给出最优收缩因子，解释高斯-马尔可夫定理的失效。
AI 中文摘要
等变性在机器学习和统计学中日益被使用，但往往缺乏系统性的论证。在一篇配套文章中，等变准则被应用于具有固定设计矩阵（固定X）的正态线性模型，在多变量不变位置-尺度群下，得到了系数向量和凝聚对角协方差矩阵的最小风险等变（MRE）估计量。我们将这些结果扩展到随机X情形，其中协变量从总体中抽样。这一扩展依赖于一个在固定X情形下是空洞的、但在随机X情形下是根本性的区别：风险和无偏性是在给定实现的设计条件下评估，还是在对设计分布取平均后评估。在条件评估下，给定X时，固定X群适用：最小二乘仍然是系数向量的最佳等变估计量，总体方差的MRE估计量保持其固定X形式，其中总体规模取实现的设计。在具有独立同分布设计的绝对评估下，图像发生质变：作用于(Y,X)的自然尺度群固定系数向量，诱导的参数空间作用是不可递的，等变风险仅在由信噪比ρ=||β||²/σ²索引的轨道上为常数，且不存在一致最小风险等变估计量。在标量情形下，最优等变权重是Oracle收缩因子w*(ρ)=ρ/(ρ+E[T⁻¹])，最小二乘在无限信号极限ρ→∞时恢复——这解释并细化了随机回归元下高斯-马尔可夫定理的已知失效。对于在位置-尺度变换下的中心化设计，最小二乘在自然不变对比类中仍然是最优的，误差方差的MRE估计量S²/(n-p+2)在两种模式下均有效。
英文摘要
Equivariance is increasingly used in machine learning and statistics, often without systematic justification. In a companion article, the equivariance criterion was applied to the normal linear model with a fixed design matrix (fixed-$X$), yielding the minimum risk equivariant (MRE) estimators of the coefficient vector and of the condensed diagonal covariance matrix under a multivariate invariant location--scale group. We extend these results to the random-$X$ case, with covariates sampled from a population. The extension hinges on a distinction vacuous for fixed-$X$ but fundamental for random-$X$: whether risk and unbiasedness are evaluated conditionally on the realized design or after averaging over the design distribution. Under conditional evaluation, the fixed-$X$ group applies given $X$: least squares remains the best equivariant estimator of the coefficient vector, and the MRE estimators of the population variances keep their fixed-$X$ forms with population sizes at the realized design. Under absolute evaluation with an i.i.d.\ design, the picture changes qualitatively: the natural scale group acting jointly on $(Y,X)$ fixes the coefficient vector, the induced parameter-space action is intransitive, equivariant risks are constant only along orbits indexed by the signal-to-noise ratio $ρ=\|β\|^2/σ^2$, and no uniformly minimum risk equivariant estimator exists. In the scalar case the optimal equivariant weight is the oracle shrinkage factor $w^*(ρ)=ρ/(ρ+E[T^{-1}])$, with least squares recovered as the infinite-signal limit $ρ\to\infty$---explaining and refining the known failure of the Gauss--Markov theorem with random regressors. For a centered design under location--scale transformations, least squares remains optimal within the natural invariant-contrast class, and the MRE estimator $S^2/(n-p+2)$ of the error variance is valid under both modes.
