多元回归中高维残差协方差矩阵的受限非线性收缩
Restricted nonlinear shrinkage of high-dimensional residual covariance matrices in multivariate regressions
AI总结：
研究非正态模型的非线性收缩协方差估计，提出无边际非线性收缩估计器MENS，给出其经验谱分布收敛性及渐近最优性，建立相变，通过模拟和回测证实理论，展示其在高维配置和决策中的实用价值。
AI中文摘要：
我们为非正态（高斯相依函数）模型开发了一种非线性收缩协方差估计理论，其中每个观测坐标是潜在高斯向量的未知严格递增变换。该模型容纳任意边际偏度和重边际尾部，同时保持高斯相依结构，是重尾、不对称金融回报的自然半参数设置。我们的估计器，即无边际非线性收缩（MENS），将一个神谕非线性收缩函数应用于正态得分秩协方差矩阵的特征值。我们给出了正态得分协方差的经验谱分布几乎必然收敛到广义马尔琴科 - 帕斯特尔定律，以及在弗罗贝尼乌斯损失下MENS在旋转不变估计器中的渐近最优性。我们为尖峰潜在相关性建立了白 - 本·阿劳斯 - 佩切相变。MENS在此类中同时实现了基于秩估计的稳健性和非线性收缩的效率。我们通过模拟研究证实了该理论，该研究分离了边际不变性属性和尖峰转变。在对标准普尔500指数股票的样本外最小方差回测中，MENS比线性收缩提供了条件更好的协方差估计、更低的实际投资组合波动率和更低的换手率，说明了其在高维配置和决策中的实用价值。
英文摘要：
We study estimation of the p*p residual scatter (shape) matrix in a high-dimensional multivariate linear regression, where p and n grow proportionally. When the coefficient matrix obeys a known linear restriction of rank q < d, as in multivariate analysis of variance, growth-curve models, and reduced-rank regression, the restricted fit leaves additional residual degrees of freedom that sharpen estimation of the shape matrix. To accommodate heavy-tailed errors, we work with independent elliptically distributed rows under a mild scale condition, a finite second moment on the radii, which is far weaker than the usual sub-Gaussian assumptions and covers every multivariate-t law with more than two degrees of freedom. Shrinking the restricted residual sample covariance directly is unsound here, since its limiting spectrum depends on the radial distribution. We instead shrink a scale-invariant scatter of the restricted residuals, whose spectrum is distribution-free over the elliptical family and obeys the same limiting law as under Gaussian errors, at a smaller effective aspect ratio. The resulting estimator attains the rotation-equivariant oracle and is asymptotically optimal within that class, and a Stein-type combination with the unrestricted estimator dominates it while remaining safe under misspecification. We further correct for the case in which the restriction is itself selected from the data. Simulations, a growth-curve experiment, and two real-data analyses illustrate the results.
