Computationally tractable bootstrap for high-dimensional data

在高维模式下，开发和研究了海量数据场景的计算易处理但完全非参数的Bootstrap。基于$n$独立的同分布$p$维观测值，其中$n$和$p$都可能是大的，我们追求结合次抽样和适当降维的创新。这种数据简化方法源于这样一种经验，即在许多情况下，每个数据的适当选择的“代表性子群”已经包含了所考虑问题的基本统计信息。对于以总体协方差矩阵的谱为特征的统计量，我们严格地引入了所谓的代表性子总体条件，并考察了它在常用统计模型中的有效性。即使在高维的情况下，这种新的方法也可以用于分布式计算和随后的平均，揭示了一种新的基于数据约简的引导方法，该方法在计算上易于处理海量数据集。

Computationally tractable but fully nonparametric bootstrap for a massive data scenario is developed and studied in the high-dimensional regime. Based on $n$ independent identically distributed $p$-dimensional observations where both, $n$ and $p$ may be large, we pursue the innovation of combining subsampling with suitable dimension reduction of the subsampled observations. This data reduction approach originates from the experience that in many situations, a suitably selected "representative subpopulation" of each datum already contains the essential statistical information for the problem under consideration. For statistics characterized by the spectrum of the population covariance matrix, we rigorously introduce the so-called representative subpopulation condition and investigate its validity in commonly used statistical models. The novel approach is accessible to distributed computation with subsequent averaging even in the high-dimensional regime, revealing a new data reduction based bootstrap which is computationally tractable for massive data sets.