Abstract: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.
| Subjects: | Methodology (stat.ME); Statistics Theory (math.ST) |
| Cite as: | arXiv:2603.16833 [stat.ME] |
| (or arXiv:2603.16833v4 [stat.ME] for this version) | |
| https://doi.org/10.48550/arXiv.2603.16833 arXiv-issued DOI via DataCite |
From: Lin Li [view email]
[v1]
Tue, 17 Mar 2026 17:39:24 UTC (21 KB)
[v2]
Mon, 23 Mar 2026 03:40:14 UTC (24 KB)
[v3]
Wed, 25 Mar 2026 18:20:59 UTC (30 KB)
[v4]
Sat, 23 May 2026 16:02:52 UTC (27 KB)
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