发表机构
School of Mathematical Sciences, Shanghai Jiao Tong University; Department of Statistics, University of Virginia; Department of Statistics and Data Science, Tsinghua University; Institute of Natural Sciences, MOE–LSC, CMA–Shanghai, SJTU–Yale Joint Center for Biostatistics and Data Science, Shanghai Jiao Tong University（上海交通大学数学科学学院; 弗吉尼亚大学统计学系; 清华大学统计与数据科学系; 上海交通大学自然科学研究院、教育部数学天元基金管理中心-上海数学中心、上海交通大学-耶鲁大学生物统计与数据科学联合中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究提出稳定的高阶估计器，无需样本分割，相比经验高阶估计器有限样本性能更稳定，且证明该类高阶估计器有类似统计保证，解决了原估计器的不足。
AI 中文摘要
高阶影响函数是构建一类统计泛函速率最优估计的统一框架，但其实践应用有限。本文提出一种无样本分割的稳定高阶估计器，其有限样本性能更稳定，且具有类似统计保证。
英文摘要
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.
CommentsThe updated version fixes a minor proof gap in the previous version
