Detection boundary in sparse regression
Detection boundary in sparse regression
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DOI：
10.1214/10-ejs589
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发表时间：
2010-01-01
影响因子：
1.1
通讯作者：
Verzelen, Nicolas
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文献类型：
--
作者：
Ingster, Yuri I.;Tsybakov, Alexandre B.;Verzelen, Nicolas
我们研究了用高斯噪声在线性回归模型中检测参数的p稀疏矢量的问题。我们建立了检测边界，即，由于样本量n和尺寸p倾向于无穷大。还展示了实现该边界的测试程序。我们的结果包括高维设置（P >> N）。主要信息是，在某些条件下，先前已为高斯序列模型建立的检测边界现象扩展到高维线性回归。最后，当噪声的方差未知时，我们建立了检测边界。有趣的是，在具有未知方差的高维设置中检测边界的速率可能与已知方差的情况不同。
We study the problem of detection of a p-dimensional sparse vector of parameters in the linear regression model with Gaussian noise. We establish the detection boundary, i.e., the necessary and sufficient conditions for the possibility of successful detection as both the sample size n and the dimension p tend to infinity. Testing procedures that achieve this boundary are also exhibited. Our results encompass the high-dimensional setting (p >> n). The main message is that, under some conditions, the detection boundary phenomenon that has been previously established for the Gaussian sequence model, extends to high-dimensional linear regression. Finally, we establish the detection boundaries when the variance of the noise is unknown. Interestingly, the rate of the detection boundary in high-dimensional setting with unknown variance can be different from the rate for the case of known variance.