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

Lin Li

Abstract

Asymptotically linear estimators in semiparametric models achieve their point-estimation guarantees via a von Mises expansion in which a second-order remainder is declared negligible. Confidence intervals then treat the first-order influence-function term as the sole source of sampling variability. This reasoning is asymptotically exact but can fail materially in finite samples whenever the second-order remainder contributes variation of the same order as the influence-function variance -- a regime we call the \emph{near-boundary regime}, characterized by nuisance estimation operating at or near the product-rate threshold. We develop a general theory of inference for this regime. Our contributions are: (i) a \emph{finite-sample variance decomposition} that separates influence-function variance from remainder-induced variance and the covariance between them; (ii) a \emph{sandwich consistency theorem} that gives a precise necessary and sufficient condition -- strong remainder negligibility -- for the standard sandwich to be consistent for the total sampling variance, and shows this is strictly stronger than the product-rate condition that guarantees asymptotic linearity; (iii) two \emph{refined variance estimators} -- leave-one-unit-out jackknife and pairs cluster bootstrap -- each with full asymptotic validity guarantees in the near-boundary regime, together with a heteroskedasticity-corrected sandwich interpretation that is numerically equivalent to the jackknife Wald interval; and (iv) a \emph{clustered-data extension} in which the remainder interacts with intra-cluster correlation to produce an analytic formula for sandwich gap amplification.

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

Abstract

Asymptotically linear estimators in semiparametric models achieve their point-estimation guarantees via a von Mises expansion in which a second-order remainder is declared negligible. Confidence intervals then treat the first-order influence-function term as the sole source of sampling variability. This reasoning is asymptotically exact but can fail materially in finite samples whenever the second-order remainder contributes variation of the same order as the influence-function variance -- a regime we call the \emph{near-boundary regime}, characterized by nuisance estimation operating at or near the product-rate threshold. We develop a general theory of inference for this regime. Our contributions are: (i) a \emph{finite-sample variance decomposition} that separates influence-function variance from remainder-induced variance and the covariance between them; (ii) a \emph{sandwich consistency theorem} that gives a precise necessary and sufficient condition -- strong remainder negligibility -- for the standard sandwich to be consistent for the total sampling variance, and shows this is strictly stronger than the product-rate condition that guarantees asymptotic linearity; (iii) two \emph{refined variance estimators} -- leave-one-unit-out jackknife and pairs cluster bootstrap -- each with full asymptotic validity guarantees in the near-boundary regime, together with a heteroskedasticity-corrected sandwich interpretation that is numerically equivalent to the jackknife Wald interval; and (iv) a \emph{clustered-data extension} in which the remainder interacts with intra-cluster correlation to produce an analytic formula for sandwich gap amplification.
Paper Structure (48 sections, 11 theorems, 53 equations, 3 tables)

This paper contains 48 sections, 11 theorems, 53 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The General Problem
  3. Motivating Application: Surrogate-Assisted Targeted Learning
  4. Relation to Existing Literature
  5. Kernel-based semiparametric estimators.
  6. Finite-dimensional nuisance estimation.
  7. Targeted variance estimation.
  8. Contributions and Outline
  9. General Framework
  10. Asymptotically Linear Estimators and the Von Mises Expansion
  11. Regularity Conditions
  12. Finite-Sample Variance Decomposition
  13. The Decomposition Theorem
  14. The Near-Boundary Gap
  15. Sandwich Consistency: A Sharp Characterization
  16. ...and 33 more sections

Key Result

Theorem 3.1

Under Assumptions ass:iid--ass:rate, the mean squared error of the ALE decomposes as: The cluster sandwich variance estimator $\widehat{\mathrm{Var}}_{\mathrm{Sand}}(\hat{\Psi})$ is consistent for $\sigma^2_{\mathrm{EIF}}/n$ only; it does not target $\mathrm{Var}(R_{\mathrm{rem}})$ or the cross term.

Theorems & Definitions (34)

  • Definition 2.1: Standard von Mises Remainder Structure
  • Definition 2.2: Product-Rate Boundary
  • Theorem 3.1: Finite-Sample Variance Decomposition
  • proof
  • Proposition 3.1: Variance Gap at the Product-Rate Boundary
  • proof
  • Remark 3.1: Constancy as a Diagnostic
  • Remark 3.2: When the Cross Term Is Non-Negligible
  • Theorem 4.1: Sandwich Consistency: Necessary and Sufficient Condition
  • proof
  • ...and 24 more