We study bootstrap inference for the $k$th largest coordinate of a normalized sum of independent high-dimensional random vectors. Existing second-order theory for maxima does not directly extend to order statistics, because the event $\{T_{n,[k]}\le t\}$ is not a rectangle and its local structure is governed by exceedance counts rather than by a single boundary. We develop an approach based on factorial moments and weighted inclusion--exclusion that reduces the problem to a collection of rare-orthant probabilities and allows high-dimensional Edgeworth and Cornish--Fisher expansions to be transferred to the order-statistic setting. Under moment, variance, and weak-dependence conditions, we derive a second-order coverage expansion for wild-bootstrap critical values of the $k$th order statistic. In particular, a third-moment matching wild bootstrap achieves coverage error of order $n^{-1}$ up to logarithmic factors, and the same second-order accuracy is obtained for a prepivoted double wild bootstrap. We also show that the maximal-correlation condition can be replaced by a stationary Gaussian exponential-mixing assumption at the price of an explicit dependence remainder $r_d$, and this remainder can itself be of order $n^{-1}$ when the dimension is sufficiently large relative to the sample size. These results extend recent second-order Gaussian and bootstrap approximation theory from maxima to the $k$th order statistic in high dimension.

翻译：我们研究独立高维随机向量归一化和的第$k$大坐标的自举推断。现有的关于最大值的二阶理论无法直接推广至顺序统计量，因为事件$\{T_{n,[k]}\le t\}$并非矩形区域，其局部结构由超越计数而非单一边界控制。我们提出一种基于阶乘矩与加权容斥原理的方法，将问题转化为一组稀有望角概率，从而允许将高维Edgeworth和Cornish-Fisher展开迁移至顺序统计量框架。在矩条件、方差条件与弱相依条件下，我们推导出第$k$阶顺序统计量野生自举临界值的二阶覆盖展开。具体而言，三阶矩匹配野生自举法可实现忽略对数因子后的$n^{-1}$阶覆盖误差，而基于预枢轴变换的双重野生自举法同样达到相同的二阶精度。我们还证明，最大相关条件可替换为平稳高斯指数混合假设，代价是引入显式相依残差$r_d$，当维度相对于样本量足够大时，该残差本身可达$n^{-1}$阶。这些结果将近期关于最大值的二阶高斯与自举逼近理论推广至高维场景下的第$k$阶顺序统计量。