论文标题
引导$ \ ell_p $ - 高维度
Bootstrapping $\ell_p$-Statistics in High Dimensions
论文作者
论文摘要
本文考虑了一种新的引导程序，以估算高维$ \ ell_p $ statistics的分布，即$ \ ell_p $ -norms $ n $ n $独立$ d $ d $ d $ d $ d $二维随机矢量，$ d \ gg n $和$ p \ in [1，1，\ infty] $。我们提供了基于高斯近似的$ \ ell_p $统计量的采样分布的非反应表征，并表明在kolmogorov-smirnov距离下，在kolmogorov-smirnov距离下，在数据的协方差结构上，引导程序是一致的。作为一般理论的应用，我们提出了一个自举假设检验，以同时推断高维平均向量。我们在高维替代方案下建立了其渐近正确性和一致性，并讨论测试的功能以及相关置信集的大小。我们在模拟数据上以数值方式说明了引导程序和测试过程。
This paper considers a new bootstrap procedure to estimate the distribution of high-dimensional $\ell_p$-statistics, i.e. the $\ell_p$-norms of the sum of $n$ independent $d$-dimensional random vectors with $d \gg n$ and $p \in [1, \infty]$. We provide a non-asymptotic characterization of the sampling distribution of $\ell_p$-statistics based on Gaussian approximation and show that the bootstrap procedure is consistent in the Kolmogorov-Smirnov distance under mild conditions on the covariance structure of the data. As an application of the general theory we propose a bootstrap hypothesis test for simultaneous inference on high-dimensional mean vectors. We establish its asymptotic correctness and consistency under high-dimensional alternatives, and discuss the power of the test as well as the size of associated confidence sets. We illustrate the bootstrap and testing procedure numerically on simulated data.
