Resampling methods are especially well-suited to inference with estimators that provide only "black-box'' access. Jackknife is a form of resampling, widely used for bias correction and variance estimation, that is well-understood under classical scaling where the sample size $n$ grows for a fixed problem. We study its behavior in application to estimating functionals using high-dimensional $Z$-estimators, allowing both the sample size $n$ and problem dimension $d$ to diverge. We begin showing that the plug-in estimator based on the $Z$-estimate suffers from a quadratic breakdown: while it is $\sqrt{n}$-consistent and asymptotically normal whenever $n \gtrsim d^2$, it fails for a broad class of problems whenever $n \lesssim d^2$. We then show that under suitable regularity conditions, applying a jackknife correction yields an estimate that is $\sqrt{n}$-consistent and asymptotically normal whenever $n\gtrsim d^{3/2}$. This provides strong motivation for the use of jackknife in high-dimensional problems where the dimension is moderate relative to sample size. We illustrate consequences of our general theory for various specific $Z$-estimators, including non-linear functionals in linear models; generalized linear models; and the inverse propensity score weighting (IPW) estimate for the average treatment effect, among others.

翻译：重采样方法特别适用于仅提供"黑盒"访问的估计量推断。刀切法作为一种重采样形式，广泛用于偏差校正和方差估计，其在样本量$n$增长而问题维度固定的经典尺度下已有充分理解。本研究探讨其在应用高维$Z$估计量估计泛函时的行为，允许样本量$n$与问题维度$d$同时发散。我们首先证明基于$Z$估计量的插件估计量存在二次崩溃现象：当$n \gtrsim d^2$时具有$\sqrt{n}$相合性与渐近正态性，但当$n \lesssim d^2$时对广泛问题类别失效。随后证明在适当正则条件下，应用刀切校正可得到当$n\gtrsim d^{3/2}$时具有$\sqrt{n}$相合性与渐近正态性的估计量。这为在维度相对于样本量处于中等水平的高维问题中使用刀切法提供了有力依据。我们通过多种具体$Z$估计量阐明一般理论的推论，包括线性模型中的非线性泛函、广义线性模型、平均处理效应的逆倾向得分加权（IPW）估计等。