Variance Inference Beyond the Sandwich for Asymptotically Linear Estimators with Second-Order Remainders

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.