This paper studies the asymptotics of resampling without replacement in the proportional regime where dimension $p$ and sample size $n$ are of the same order. For a given dataset $(\bm{X},\bm{y})\in\mathbb{R}^{n\times p}\times \mathbb{R}^n$ and fixed subsample ratio $q\in(0,1)$, the practitioner samples independently of $(\bm{X},\bm{y})$ iid subsets $I_1,...,I_M$ of $\{1,...,n\}$ of size $q n$ and trains estimators $\bm{\hat{\beta}}(I_1),...,\bm{\hat{\beta}}(I_M)$ on the corresponding subsets of rows of $(\bm{X},\bm{y})$. Understanding the performance of the bagged estimate $\bm{\bar{\beta}} = \frac1M\sum_{m=1}^M \bm{\hat{\beta}}(I_1),...,\bm{\hat{\beta}}(I_M)$, for instance its squared error, requires us to understand correlations between two distinct $\bm{\hat{\beta}}(I_m)$ and $\bm{\hat{\beta}}(I_{m'})$ trained on different subsets $I_m$ and $I_{m'}$. In robust linear regression and logistic regression, we characterize the limit in probability of the correlation between two estimates trained on different subsets of the data. The limit is characterized as the unique solution of a simple nonlinear equation. We further provide data-driven estimators that are consistent for estimating this limit. These estimators of the limiting correlation allow us to estimate the squared error of the bagged estimate $\bm{\bar{\beta}}$, and for instance perform parameter tuning to choose the optimal subsample ratio $q$. As a by-product of the proof argument, we obtain the limiting distribution of the bivariate pair $(\bm{x}_i^T \bm{\hat{\beta}}(I_m), \bm{x}_i^T \bm{\hat{\beta}}(I_{m'}))$ for observations $i\in I_m\cap I_{m'}$, i.e., for observations used to train both estimates.

翻译：本文研究在维度$p$与样本量$n$同阶的比例机制下，无放回重抽样的渐近性质。给定数据集$(\bm{X},\bm{y})\in\mathbb{R}^{n\times p}\times \mathbb{R}^n$和固定子样本比例$q\in(0,1)$，从业者独立于$(\bm{X},\bm{y})$抽样$\{1,...,n\}$的独立同分布子集$I_1,...,I_M$，每个子集大小为$q n$，并基于$(\bm{X},\bm{y})$的对应行子集训练估计量$\bm{\hat{\beta}}(I_1),...,\bm{\hat{\beta}}(I_M)$。理解袋装估计量$\bm{\bar{\beta}} = \frac1M\sum_{m=1}^M \bm{\hat{\beta}}(I_1),...,\bm{\hat{\beta}}(I_M)$的性能（例如其平方误差），需要理解训练于不同子集$I_m$和$I_{m'}$的两个不同$\bm{\hat{\beta}}(I_m)$与$\bm{\hat{\beta}}(I_{m'})$之间的相关性。在鲁棒线性回归和逻辑回归中，我们刻画了训练于不同数据子集的两个估计量之间相关性的概率极限。该极限被表征为某个简单非线性方程的唯一解。我们进一步提供了对此极限进行一致估计的数据驱动估计量。这些极限相关性的估计量使我们能够估计袋装估计量$\bm{\bar{\beta}}$的平方误差，并例如通过参数调优选择最优子样本比例$q$。作为证明过程的副产品，我们得到了对于观测值$i\in I_m\cap I_{m'}$（即用于训练两个估计量的观测值）的二元对$(\bm{x}_i^T \bm{\hat{\beta}}(I_m), \bm{x}_i^T \bm{\hat{\beta}}(I_{m'}))$的极限分布。