Refined Inference for Asymptotically Linear Estimators with Non-Negligible Second-Order Remainders

TL;DR

This paper investigates the limitations of traditional variance estimation methods for semiparametric estimators in the near-boundary regime, proposing refined variance estimation techniques that improve inference accuracy.

Contribution

It introduces a finite-sample variance decomposition, characterizes when sandwich variance underestimates total variance, and proposes jackknife and bootstrap methods for better variance estimation.

Findings

The variance ratio $ ext{V}_{ ext{JK}}/ ext{V}_{ ext{Sand}}$ ranges from 1.14 to 1.38 in simulations.

Refined procedures significantly improve coverage in clustered data settings.

Jackknife and bootstrap methods can consistently estimate total variance under certain regularity conditions.

Abstract

Semiparametric estimators admitting a von Mises expansion often reduce inference to the influence-function variance. This reduction is justified when the second-order remainder is negligible in variance, a condition that is stronger than the usual product-rate requirement guaranteeing classical asymptotic linearity. When the remainder contributes non-negligible variance, the standard sandwich can underestimate the total sampling variance and Wald intervals can undercover; we call this the \emph{near-boundary regime}. We derive a finite-sample variance decomposition separating influence-function and remainder components, give a practical characterization of when sandwich variance can fail, and show that the leave-one-out jackknife and pairs cluster bootstrap can estimate the total variance under explicit regularity conditions. For the jackknife, consistency follows from a self-normalization argument; for the bootstrap, we work under a Mallows-2 consistency condition. An analytic expression for the amplification of the sandwich gap by intra-cluster correlation is derived for clustered data. A simulation study using a surrogate-assisted targeted learning estimator in stepped-wedge cluster-randomized trials illustrates the regime: the variance ratio $\hat{V}_{\rm JK}/\hat{V}_{\rm Sand}$ is 1.14--1.38 and persistent across cluster counts, and the refined procedures substantially improve coverage.