高维线性模型中参数组的显著性检验

Significance Tests for Parameter Group in High-Dimensional Linear Models

摘要: 高维线性模型中参数的显著性检验是统计学研究的热点。现有的统计方法往往局限于参数的稀疏性假设，但在实际中，该假设是很难检验且极容易被违反的。本文通过重构回归对参数组进行显著性检验，首先利用原假设的结构得到重构回归模型，并建立设计矩阵间的线性相关模型，进而计算两模型误差的相关性，以此将待测原假设转化为可检验的矩条件，并建立检验统计量，得到检验统计量的渐近分布。最后，通过模拟试验，观察该检验方法在不同的误差和设计矩阵设置下的第I类错误概率和势函数曲线。试验结果表明：对于参数组检验，矩方法能保证第I类错误概率控制在显著性水平附近，并且可以证明检验的势特征。

Abstract: The significance test of parameters in high-dimensional linear models is a hot topic in statistical re-search. Existing statistical methods are often limited to the assumption of parameter sparsity, but in practice, this assumption is difficult to test and easily violated. In this paper, the significance test of the parameter group is carried out through reconstruction regression. First, the reconstruction regression model is obtained by using the structure of the null hypothesis, and the linear correla-tion model between the design matrices is established, and then the correlation between the errors of the two models is calculated. The hypothesis is transformed into a testable moment condition, establish the test statistic, and obtain the asymptotic distribution of the test statistic. Finally, through simulation experiments, the probability of Type I error of the test method and power func-tion curve under different error and design matrix settings are observed. The experimental results show that for the parameter group test, the moment method can ensure that the probability of Type I error is close to the significance level, and can prove the power property of the test.