Testing in high-dimensional spiked models
Testing in high-dimensional spiked models
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DOI：
10.1214/18-aos1697
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发表时间：
2015-09
期刊：
影响因子：
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通讯作者：
I. Johnstone;A. Onatski
中科院分区：
文献类型：
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作者：
I. Johnstone;A. Onatski
我们考虑了James(1964)发现的五类多元统计问题，这些问题一起涵盖了经典多变量分析的大部分，加上一个更简单的极限情况，对称矩阵去噪。每个James问题都涉及$E^{-1}H$的特征值，其中$H$和$E$与高维Wishart矩阵成正比。在零假设下，两种心愿都是同一性协方差的中心。在另一种情况下，$H$的非中心性或协方差参数有一个单独的特征值，即尖峰。当尖峰小于特定于案例的相变阈值时，没有一个样本特征值与整体分离，这使得测试问题具有挑战性。对于这六种情况，我们使用一个统一的策略，证明了以次临界尖峰的值为参数的对数似然比过程收敛于具有对数相关性的高斯过程。然后，我们推导出用于尖峰存在检验的渐近功率包络。
We consider the five classes of multivariate statistical problems identified by James (1964), which together cover much of classical multivariate analysis, plus a simpler limiting case, symmetric matrix denoising. Each of James' problems involves the eigenvalues of $E^{-1}H$ where $H$ and $E$ are proportional to high dimensional Wishart matrices. Under the null hypothesis, both Wisharts are central with identity covariance. Under the alternative, the non-centrality or the covariance parameter of $H$ has a single eigenvalue, a spike, that stands alone. When the spike is smaller than a case-specific phase transition threshold, none of the sample eigenvalues separate from the bulk, making the testing problem challenging. Using a unified strategy for the six cases, we show that the log likelihood ratio processes parameterized by the value of the sub-critical spike converge to Gaussian processes with logarithmic correlation. We then derive asymptotic power envelopes for tests for the presence of a spike.