标题： 高维线性模型中的残差方差和信噪比 显示英文标题

标题： Residual variance and the signal-to-noise ratio in high-dimensional linear models

摘要： 残差方差和信噪比是许多统计模型和模型拟合过程中的重要量。它们在回归诊断中起着重要作用，并且在确定估计和预测问题的性能极限以及在许多流行的高维数据分析正则化回归方法中的收缩参数选择中也至关重要。我们提出了新的残差方差、l2-信号强度和信噪比的估计器，这些估计器在具有高维线性模型的高维线性模型中是一致的并且渐近正态分布的，其中预测变量的数量d与观察值的数量n成比例。 现有关于高维线性模型中残差方差估计的结果依赖于潜在信号的稀疏性。我们的结果不需要稀疏性假设，并表明即使当d > n且潜在信号本身不可估计时，残差方差也可能被一致地估计。基本数值工作表明，我们理论结果所作的一些分布假设可能可以放宽。 显示英文摘要

摘要： Residual variance and the signal-to-noise ratio are important quantities in many statistical models and model fitting procedures. They play an important role in regression diagnostics, in determining the performance limits in estimation and prediction problems, and in shrinkage parameter selection in many popular regularized regression methods for high-dimensional data analysis. We propose new estimators for the residual variance, the l2-signal strength, and the signal-to-noise ratio that are consistent and asymptotically normal in high-dimensional linear models with Gaussian predictors and errors, where the number of predictors d is proportional to the number of observations n. Existing results on residual variance estimation in high-dimensional linear models depend on sparsity in the underlying signal. Our results require no sparsity assumptions and imply that the residual variance may be consistently estimated even when d > n and the underlying signal itself is non-estimable. Basic numerical work suggests that some of the distributional assumptions made for our theoretical results may be relaxed.