The standard recipe for semiparametric inference: expand the estimator via von Mises expansion, identify the linear influence function, declare the second-order remainder negligible, construct confidence intervals from the influence function's variance.
The declaration is the problem. In finite samples, the second-order remainder can contribute variance of the same order as the influence function itself. When it does, the standard confidence intervals are miscalibrated — not because the theory is wrong, but because the theory's regime assumption (remainder negligible) fails to hold.
Li and Wu call this the near-boundary regime: the estimator is asymptotically linear, the expansion is valid, but the convergence rate is such that the remainder matters at practical sample sizes. The standard sandwich variance estimator, which captures only the influence function contribution, systematically underestimates uncertainty.
Their fix is a finite-sample variance decomposition that explicitly separates influence-function variance from remainder-induced variance. When the remainder contribution is small, the standard approach is recovered. When it is not, jackknife or pairs cluster bootstrap estimators provide valid coverage.
The structural lesson: “asymptotically negligible” is a statement about limits, not about samples. Every practicing statistician knows this informally — small-sample behavior differs from asymptotic guarantees. What this paper formalizes is the precise condition under which the difference matters, and provides estimators that adapt to it. The negligible remainder is negligible only when you check.