Computationally tractable bootstrap for high-dimensional data

在高维环境下，研究了一种计算上易于处理但完全非参数的海量数据自举方法。基于$n$独立同分布的$p$维观测值，其中$n$和$p$可能都很大，我们追求将子抽样与适当的子抽样观测值降维相结合的创新。这种减少数据的办法源于这样一种经验，即在许多情况下，每个基准中适当选择的“代表性亚群”已经包含了所审议问题的基本统计资料。对于以总体协方差矩阵谱为特征的统计量，我们严格地引入了所谓的代表性亚种群条件，并研究了其在常用统计模型中的有效性。该方法可用于分布式计算，即使在高维状态下也可进行后续平均，揭示了一种新的基于数据约简的自举方法，该方法在计算上可处理大量数据集。

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.