论文标题
渐近贝叶斯在稀疏下的优化性，用于等超相关的多元正常测试统计
Asymptotic bayes optimality under sparsity for equicorrelated multivariate normal test statistics
论文作者
论文摘要
在这里，我们解决了在存在稀疏替代方案的情况下与渐近贝叶斯的最佳测试有关的测试统计数据之间的依赖性。扩展Bogdan等人的设置。 （2011年）我们考虑了对测试统计的联合分布的超相关（具有相等相关$ρ$）的多变量正常假设，而在平均值矢量$ \boldsymbolμ$上进行条件。该设置的其余部分与Bogdan等人相同。 （2011）在渐近框架中有轻微的修改。我们利用具有相等边缘方差的等效相关的多元正常变量来利用众所周知的结果，以将测试统计量分解为独立的随机变量。然后，我们确定了一组独立但不可观察的高斯随机变量，足以解决多个测试问题，并根据Bogdan等人的那些虚拟变量来降低单个截止测试的必要条件，以使单个临界测试成为ABO。 （2011）。此外，我们用统计数据偏离了算术手段来代替虚拟变量，这些变量很容易根据早期使用的分解而易于计算。然后得出其他假设，以便使用独立的虚拟变量也可以在替换变量的情况下使用独立的虚拟变量的单个临界测试的必要条件（且差额$ o（1）$）。接下来，具有相同的额外假设，单个截止测试的必要条件是控制贝叶斯FDR的必要条件，因此，如果满足相同的条件，则在各种稀疏性假设下，我们证明了多个测试的经典Bonferroni和Benjamini-Hochberg方法是ABOS。
Here we address dependence among the test statistics in connection with asymptotically Bayes' optimal tests in presence of sparse alternatives. Extending the setup in Bogdan et.al. (2011) we consider an equicorrelated ( with equal correlation $ρ$ ) multivariate normal assumption on the joint distribution of the test statistics, while conditioned on the mean vector $\boldsymbolμ$. Rest of the set up is identical to Bogdan et.al. (2011) with a slight modification in the asymptotic framework. We exploit an well known result on equicorrelated multivariate normal variables with equal marginal variances to decompose the test statistics into independent random variables. We then identify a set of independent yet unobservable gaussian random variables sufficient for the multiple testing problem and chalk out the necessary and sufficient conditions for single cutoff tests to be ABOS based on those dummy variables following Bogdan et.al. (2011). Further we replaced the dummy variables with deviations of the statistics from their arithmetic means which were easily calculable from the observations due to the decomposition used earlier. Additional assumptions are then derived so that the necessary and sufficient conditions for single cutoff tests to be ABOS using the independent dummy variables plays the same role with the replacement variable as well (with a deviation of order $o(1)$). Next with the same additional assumption, necessary and sufficient conditions for single cutoff tests to control the Bayesian FDRs are derived and as a consequence under various sparsity assumptions we proved that the classical Bonferroni and Benjamini-Hochberg methods of multiple testing are ABOS if the same conditions are satisfied.
