Stabilized Higher-Order Influence Functions: Statistical Theory of a Class of Bilinear Forms
稳定化高阶影响函数：一类双线性形式的统计理论AI 解读 ↓
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稳定化高阶影响函数：一类双线性形式的统计理论
- 研究问题
- 高阶影响函数估计依赖高维密度或样本 Gram 矩阵求逆而面临计算不稳定时，如何构造更稳定的估计量并保留相近统计保证？
- 方法与思路
- 作者针对一类双线性形式或泛函，提出无需样本分割的稳定化高阶估计量，并比较其与使用样本 Gram 矩阵求逆的经验高阶估计量。
- 材料陈述的贡献
- 摘要称新估计量相较经验高阶估计量具有更稳定的有限样本表现，并证明这一类稳定化估计量享有类似的统计理论保证。
- 阅读关注点
- 应核对稳定化的具体构造、双线性泛函范围、基函数维数与样本量的关系，以及取消样本分割后如何控制依赖、偏差和方差。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未核实；摘要未列出类似统计保证的准确形式，也未量化稳定性改善或证明其普遍优越。
原文摘要
Higher-order influence functions, introduced in a series of articles (Robins et al., 2008, 2009a; van der Vaart, 2014; Robins et al., 2016, 2023; Liu et al., 2017), are a unified framework for constructing rate-optimal point estimates of a class of statistical functionals under various complexity-reducing assumptions on the posited statistical model that generates the observed data. Although higher-order (influence functions) estimators are theoretically appealing, they have very limited practical uptake compared to their first-order counterparts. The original higher-order estimators proposed in Robins et al. (2008) and Robins et al. (2017) involve nonparametric density estimation of multi-dimensional covariates, a highly nontrivial statistical and computational problem on its own. The density estimator is, in turn, used in the evaluation of the inverse population Gram matrix $Ω$ of a set of $k$-dimensional basis transformations of covariates. There, $k$ is allowed to be as large as $o (n^2)$. To partially address this potential shortcoming, Liu et al. (2017) restrict $k$ to $o (n)$ and instead estimate $Ω$ directly using the inverse sample Gram matrix estimator, but computed from an independent sample often obtained by sample-splitting. Liu et al. (2017) refer to this alternative estimator as the empirical higher-order estimator. Although the empirical higher-order estimator bypasses density estimation, it suffers from numerical instability due to inverting a large-dimensional sample Gram matrix. In this article, for a class of bilinear forms/functionals that often appear in substantive fields, we propose a new stabilized higher-order estimator without sample splitting, which exhibits more stable finite-sample performance compared to the empirical higher-order estimator. We also prove that this new class of higher-order estimators enjoys similar statistical guarantees.