HYPOTHESIS TESTING FOR HIGH-DIMENSIONAL SPARSE BINARY REGRESSION.
HYPOTHESIS TESTING FOR HIGH-DIMENSIONAL SPARSE BINARY REGRESSION.
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DOI：
10.1214/14-aos1279
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发表时间：
2015-02
影响因子：
4.5
通讯作者：
Lin X
中科院分区：
文献类型：
--
作者：
Mukherjee R;Pillai NS;Lin X
在本文中，我们研究了高维稀疏二元回归模型中极小极大假设检验的检测边界。受罕见变异效应基因测序关联研究的启发，我们研究了设计矩阵稀疏时假设检验问题的复杂性。我们观察到检测边界行为中的一个新现象，这在高斯线性回归的情况下不会发生。我们将检测边界导出为两个分量的函数：设计矩阵稀疏指数和信号强度，其中每个分量都是替代方案稀疏性的函数。对于任何替代方案，如果设计矩阵稀疏指数太高，则无论信号强度大小如何，任何测试都渐近无能为力。对于稀疏指数不太高的二元设计矩阵，我们的结果与高斯情况下的结果相似。在这种情况下，我们得出密集和稀疏区域的检测边界。对于密集状态，我们证明广义似然比是速率最优的；对于稀疏机制，我们提出了扩展的更高批评测试，并表明它是速率最优且尖锐的。我们使用模拟研究来说明理论结果的有限样本属性。
In this paper, we study the detection boundary for minimax hypothesis testing in the context of high-dimensional, sparse binary regression models. Motivated by genetic sequencing association studies for rare variant effects, we investigate the complexity of the hypothesis testing problem when the design matrix is sparse. We observe a new phenomenon in the behavior of detection boundary which does not occur in the case of Gaussian linear regression. We derive the detection boundary as a function of two components: a design matrix sparsity index and signal strength, each of which is a function of the sparsity of the alternative. For any alternative, if the design matrix sparsity index is too high, any test is asymptotically powerless irrespective of the magnitude of signal strength. For binary design matrices with the sparsity index that is not too high, our results are parallel to those in the Gaussian case. In this context, we derive detection boundaries for both dense and sparse regimes. For the dense regime, we show that the generalized likelihood ratio is rate optimal; for the sparse regime, we propose an extended Higher Criticism Test and show it is rate optimal and sharp. We illustrate the finite sample properties of the theoretical results using simulation studies.