🤖 AI Summary
This work addresses the computational and numerical challenges that commonly arise in practical implementations of higher-order influence function estimation, which often suffer from high-dimensional density estimation or inversion of large Gram matrices. The authors propose a stabilized estimation procedure that eliminates the need for sample splitting by incorporating Gram matrix regularization and a bilinear form structure. This approach avoids high-dimensional density estimation altogether while substantially improving numerical stability in finite samples. The method retains the theoretically optimal convergence rate and provides strong statistical guarantees alongside robust empirical performance, effectively overcoming key limitations of existing higher-order influence function estimators.
📝 Abstract
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 potentially inverting a large-dimensional sample Gram matrix. In this article, 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, and more importantly, we prove that this new class of higher-order estimators enjoys similar statistical guarantees.