Gaussian Multiplier Bootstrap Procedure for the $κ$th Largest Coordinate of High-Dimensional Statistics

We consider the problem of Gaussian and bootstrap approximations for the distribution of the $κ$th-largest statistic in high dimensions. This statistic, defined as the $κ$th-largest component of the sum of independent random vectors, is critical in numerous high-dimensional estimation and testing problems. Such a problem has been studied previously for $κ=1$ (i.e., maxima). However, in many applications, a general $κ\geq1$ is of great interest, which is addressed in this paper. By invoking the iterative randomized Lindeberg method, we provide bounds for the errors in distributional approximations. These bounds generalize existing results and extend the applicability to a wider range of bootstrap methods. All these results allow the dimension $p$ of random vectors to be as large as or much larger than the sample size $n$. Extensive simulation results and real data analysis demonstrate the effectiveness and advantages of the Gaussian multiplier bootstrap procedure.