发表机构
Department of Applied Economics and Statistics, University of Delaware（应用经济学与统计学系，德雷克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究主成分回归中秩选择不足问题，提出去底主成分回归方法，证明其预测风险上下限，通过风险分解解释分离现象，给出同一样本截尾均值下限估计及相关公式，表明该方法改进秩为1的PCR，对马尔琴科 - 帕斯特尔主体平均下限减法非最优。
AI 中文摘要
主成分回归（PCR）通过选择谱截止来正则化高维预测，但秩选择无法校正保留的经验特征值的系统性膨胀。我们研究了干净的高斯随机设计，其中总体协方差尾部产生了一个与预测头部尺度相当的几乎标量的样本空间下限。去底主成分回归（dPCR）保留截止值，并从保留的分母中减去估计的下限。我们证明了在所有秩上的普通PCR预测风险下限和高概率的dPCR上限。当在总体预测风险中下限明显且去除成本低时，相对于最佳普通PCR秩，dPCR的条件风险渐近可忽略不计。精确的风险分解解释了这种分离：分母膨胀由第一谱质量控制，而校正的干净预测成本由平方谱质量控制。同一样本的截尾均值下限估计在预定秩处达到最优dPCR上限率，并且当尾部预测能量分数消失时，在近似预测对齐下分离仍然存在。单独的逐点固定方面公式表明，风险最优的正标量校正改进了秩为1的PCR，而对于广泛的马尔琴科 - 帕斯特尔主体，平均下限减法通常不是最优的。
英文摘要
Principal component regression (PCR) regularizes high-dimensional prediction by choosing a spectral cutoff, but rank selection cannot correct systematic inflation of the retained empirical eigenvalues. We study clean Gaussian random designs in which the aggregate covariance tail creates a nearly scalar sample-space floor comparable to the predictive head scale. De-floored principal component regression (dPCR) retains the cutoff and subtracts an estimated floor from the retained denominators. We prove an ordinary-PCR prediction-risk lower bound uniform over all ranks and a high-probability dPCR upper bound. When the floor is sharp and inexpensive to remove in population prediction risk, the conditional risk of dPCR is asymptotically negligible relative to that of the best ordinary PCR rank. An exact risk decomposition explains the separation: denominator inflation is governed by first spectral mass, whereas the clean prediction cost of correction is governed by squared spectral mass. A same-sample trimmed-mean floor estimate attains the oracle dPCR upper-bound rate at a prespecified rank, and the separation persists under approximate predictive alignment when the tail prediction-energy fraction vanishes. Separate pointwise fixed-aspect formulas show that the risk-optimal positive scalar correction improves rank-$1$ PCR, whereas mean-floor subtraction is generally not optimal for a broad Marchenko--Pastur bulk.
