In this article, we propose a class of test statistics for a change point in the mean of high-dimensional independent data. Our test integrates the U-statistic based approach in a recent work by \cite{hdcp} and the $L_q$-norm based high-dimensional test in \cite{he2018}, and inherits several appealing features such as being tuning parameter free and asymptotic independence for test statistics corresponding to even $q$s. A simple combination of test statistics corresponding to several different $q$s leads to a test with adaptive power property, that is, it can be powerful against both sparse and dense alternatives. On the estimation front, we obtain the convergence rate of the maximizer of our test statistic standardized by sample size when there is one change-point in mean and $q=2$, and propose to combine our tests with a wild binary segmentation (WBS) algorithm to estimate the change-point number and locations when there are multiple change-points. Numerical comparisons using both simulated and real data demonstrate the advantage of our adaptive test and its corresponding estimation method.
翻译:在本篇文章中,我们为高维独立数据平均值的变化点建议了一组测试统计数据。我们的测试将U-统计基础方法纳入到\ cite{hdcp}和$_q$-norm基于在\cite{he2018}中的高维测试的近期工作中,并继承了几个吸引人的特征,例如对参数进行免费调试,对测试统计数据进行无线和无线独立,与美元相对应。与若干不同的美元相对应的测试统计数据简单结合,导致对适应性能量属性进行测试,也就是说,对稀有和密集的替代品都具有强大的影响力。在估算前,我们获得了我们测试统计数据最大化的趋同率,按样本大小标准化,当平均值有一个变化点和$q=2美元时,我们建议将我们的测试与野生二分解算法结合起来,以估计变化点数和多点时的位置。使用模拟数据和实际数据进行的数字比较,显示了我们适应性测试及其相应估算方法的优势。