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Semiparametric Inference under Dual Positivity Boundaries:Nested Identification with Administrative Censoring and Confounded Treatment

Lin Li

Abstract

When a long-term outcome is administratively censored for a substantial fraction of a study cohort while a short-term intermediate variable remains broadly available, the target causal parameter can be identified through a nested functional that integrates the outcome regression over the conditional intermediate distribution, avoiding inverse censoring weights entirely. In observational studies where treatment is also confounded, this nested identification creates a semiparametric structure with two distinct positivity boundaries -- one from the censoring mechanism and one from the treatment assignment -- that enter the efficient influence function in fundamentally different roles. The censoring boundary is removed from the identification by the nested functional but remains in the efficient score; the treatment boundary appears in both. We develop the inference theory for this dual-boundary structure. Three results are established.

Semiparametric Inference under Dual Positivity Boundaries:Nested Identification with Administrative Censoring and Confounded Treatment

Abstract

When a long-term outcome is administratively censored for a substantial fraction of a study cohort while a short-term intermediate variable remains broadly available, the target causal parameter can be identified through a nested functional that integrates the outcome regression over the conditional intermediate distribution, avoiding inverse censoring weights entirely. In observational studies where treatment is also confounded, this nested identification creates a semiparametric structure with two distinct positivity boundaries -- one from the censoring mechanism and one from the treatment assignment -- that enter the efficient influence function in fundamentally different roles. The censoring boundary is removed from the identification by the nested functional but remains in the efficient score; the treatment boundary appears in both. We develop the inference theory for this dual-boundary structure. Three results are established.
Paper Structure (26 sections, 9 theorems, 8 equations, 4 tables)

This paper contains 26 sections, 9 theorems, 8 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Nested Identification under Two Positivity Boundaries
  3. Observed-data structure
  4. Conditions defining the dual-boundary structure
  5. Identification
  6. Semiparametric Theory
  7. Efficient influence function
  8. Remainder decomposition
  9. Fluctuation coupling in nested functionals
  10. Three-component robustness geometry
  11. Cross-fitting
  12. Dual near-boundary variance decomposition
  13. Targeted Estimator Construction
  14. Asymptotic Theory
  15. Simulation Study
  16. ...and 11 more sections

Key Result

Theorem 1

Under Assumptions a:consistency--a:overlap: where $\bar{Q}_Y(S,a,W) = E[Y \mid S, A\!=\!a, W, \Delta\!=\!1]$.

Theorems & Definitions (19)

  • Theorem 1: Identification via the nested functional
  • proof
  • Remark 1: What the nested functional separates
  • Lemma 2: Algebraic elimination of $R_{SY}$
  • Proposition 3: Fluctuation coupling
  • Remark 2: The structural role of conditional mean zero
  • Remark 3: What propensity misspecification costs
  • Theorem 4: Product-rate conditions for the dual-boundary functional
  • Corollary 5: Sufficient configurations and their regimes
  • Remark 4: Contrast with classical and multiply robust structures
  • ...and 9 more