Estimating the error variance in a high-dimensional linear model
Estimating the error variance in a high-dimensional linear model
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DOI：
10.1093/biomet/asz017
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发表时间：
2017-12
期刊：
影响因子：
2.7
通讯作者：
Guo Yu;J. Bien
中科院分区：
文献类型：
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作者：
Guo Yu;J. Bien
已广泛研究了套索作为估计高维线性模型中系数矢量的工具。但是，在此上下文中估计误差差异知之甚少。在本文中，我们提出了误差差异的天然拉索估计量，从而最大程度地提高了受惩罚的可能性目标。自然套索的一个关键方面是，可能性是根据具有未知平均值和方差的高斯的多参数指数家族的自然参数化表示。结果是对误差差异的一个非常简单的估计器，就均方误差而言，证明性能良好。这些理论结果不需要在设计矩阵或真实回归系数上放置任何假设。我们还提出了一个称为有机拉索的伴侣估计量，从理论上讲，该估计量不需要调整正则化参数。与先前存在的方法相比，这两个估计器在经验上做得很好，尤其是在成功恢复系数向量的真正支持的环境中。最后，我们表明，现有方法在比以前所知的假设少的情况下可以做得很好，从而提供了一个关于估计高维线性模型中误差差异的问题的完整故事。
The lasso has been studied extensively as a tool for estimating the coefficient vector in the high-dimensional linear model; however, considerably less is known about estimating the error variance in this context. In this paper, we propose the natural lasso estimator for the error variance, which maximizes a penalized likelihood objective. A key aspect of the natural lasso is that the likelihood is expressed in terms of the natural parameterization of the multi-parameter exponential family of a Gaussian with unknown mean and variance. The result is a remarkably simple estimator of the error variance with provably good performance in terms of mean squared error. These theoretical results do not require placing any assumptions on the design matrix or the true regression coefficients. We also propose a companion estimator, called the organic lasso, which theoretically does not require tuning of the regularization parameter. Both estimators do well empirically compared to pre-existing methods, especially in settings where successful recovery of the true support of the coefficient vector is hard. Finally, we show that existing methods can do well under fewer assumptions than previously known, thus providing a fuller story about the problem of estimating the error variance in high-dimensional linear models.