Confidence intervals for low dimensional parameters in high dimensional linear models
Confidence intervals for low dimensional parameters in high dimensional linear models
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DOI：
10.1111/rssb.12026
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发表时间：
2014-01-01
影响因子：
5.8
通讯作者：
Zhang, Stephanie S.
中科院分区：
文献类型：
--
作者：
Zhang, Cun-Hui;Zhang, Stephanie S.
本文的目的是提出方法的统计推断的低维参数与高维数据。我们专注于构建线性回归模型中单个系数和其中几个系数的线性组合的置信区间，尽管我们的想法适用于更广泛的背景。所提出的理论结果提供了所提出的估计量的渐近正态性沿着与其有限维协方差矩阵的一致估计量的充分条件。这些充分条件允许变量的数量超过样本大小和存在许多小的非零系数。我们的方法和理论适用于一个预先设想的回归系数或对比的区间估计以及多个回归系数的同时区间估计。此外，该方法提出的回归数据变成一个近似的高斯序列的点估计的个别回归系数，它可以用来选择变量后，适当的阈值。仿真结果表明，所提出的置信区间的覆盖概率的准确性以及其他理想的属性，有力地支持了理论结果。
The purpose of this paper is to propose methodologies for statistical inference of low dimensional parameters with high dimensional data. We focus on constructing confidence intervals for individual coefficients and linear combinations of several of them in a linear regression model, although our ideas are applicable in a much broader context. The theoretical results that are presented provide sufficient conditions for the asymptotic normality of the proposed estimators along with a consistent estimator for their finite dimensional covariance matrices. These sufficient conditions allow the number of variables to exceed the sample size and the presence of many small non-zero coefficients. Our methods and theory apply to interval estimation of a preconceived regression coefficient or contrast as well as simultaneous interval estimation of many regression coefficients. Moreover, the method proposed turns the regression data into an approximate Gaussian sequence of point estimators of individual regression coefficients, which can be used to select variables after proper thresholding. The simulation results that are presented demonstrate the accuracy of the coverage probability of the confidence intervals proposed as well as other desirable properties, strongly supporting the theoretical results.