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

Asymptotically linear estimators in semiparametric models are usually studied through a von Mises expansion in which first-order inference is based on the influence-function variance. This reduction is valid only when the second-order remainder is negligible not only in probability but also in variance, a requirement not implied by the usual product-rate conditions ensuring asymptotic linearity. We study the regime in which the remainder contributes variance at order $n^{-1}$, so that the total sampling variance differs from the standard influence-function approximation by a non-vanishing first-order term. We derive a finite-sample variance decomposition separating the influence-function variance, the remainder variance, and their covariance, and characterize sandwich validity through the vanishing of scaled remainder variance: under a negligible cross term, the sandwich estimator is consistent for the total sampling variance when $n\,\mathrm{Var}(R_{\mathrm{rem}})\to 0$ and materially underestimates it in the complementary near-boundary regime $n\,\mathrm{Var}(R_{\mathrm{rem}})\to c_R>0$. We then establish asymptotic validity of two refined procedures in the near-boundary regime: the leave-one-out jackknife and the pairs bootstrap. Jackknife validity is obtained through a self-normalization argument; bootstrap validity is established directly under a Mallows--2 condition. We also extend the theory to clustered data and derive an analytic expression showing how intra-cluster correlation amplifies the sandwich gap through the remainder term. Simulations illustrate the regime and confirm the predicted coverage behaviour of the competing variance estimators.