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Kernel Debiased Plug-in Estimation based on the Universal Least Favorable Submodel

Haiyi Chen, Yang Liu, Ivana Malenica

Abstract

We propose ULFS-KDPE, a kernel debiased plug-in estimator based on the universal least favorable submodel, for estimating pathwise differentiable parameters in nonparametric models. The method constructs a data-adaptive debiasing flow in a reproducing kernel Hilbert space (RKHS), producing a plug-in estimator that achieves semiparametric efficiency without requiring explicit derivation or evaluation of efficient influence functions. We place ULFS-KDPE on a rigorous functional-analytic foundation by formulating the universal least favorable update as a nonlinear ordinary differential equation on probability densities. We establish existence, uniqueness, stability, and finite-time convergence of the empirical score along the induced flow. Under standard regularity conditions, the resulting estimator is regular, asymptotically linear, and attains the semiparametric efficiency bound simultaneously for a broad class of pathwise differentiable parameters. The method admits a computationally tractable implementation based on finite-dimensional kernel representations and principled stopping criteria. In finite samples, the combination of solving a rich collection of score equations with RKHS-based smoothing and avoidance of direct influence-function evaluation leads to improved numerical stability. Simulation studies illustrate the method and support the theoretical results.

Kernel Debiased Plug-in Estimation based on the Universal Least Favorable Submodel

Abstract

We propose ULFS-KDPE, a kernel debiased plug-in estimator based on the universal least favorable submodel, for estimating pathwise differentiable parameters in nonparametric models. The method constructs a data-adaptive debiasing flow in a reproducing kernel Hilbert space (RKHS), producing a plug-in estimator that achieves semiparametric efficiency without requiring explicit derivation or evaluation of efficient influence functions. We place ULFS-KDPE on a rigorous functional-analytic foundation by formulating the universal least favorable update as a nonlinear ordinary differential equation on probability densities. We establish existence, uniqueness, stability, and finite-time convergence of the empirical score along the induced flow. Under standard regularity conditions, the resulting estimator is regular, asymptotically linear, and attains the semiparametric efficiency bound simultaneously for a broad class of pathwise differentiable parameters. The method admits a computationally tractable implementation based on finite-dimensional kernel representations and principled stopping criteria. In finite samples, the combination of solving a rich collection of score equations with RKHS-based smoothing and avoidance of direct influence-function evaluation leads to improved numerical stability. Simulation studies illustrate the method and support the theoretical results.
Paper Structure (48 sections, 28 theorems, 190 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 48 sections, 28 theorems, 190 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Problem Setup and Notation
  5. Target Parameter
  6. Preliminaries
  7. Scores
  8. Influence function
  9. Least favorable submodels
  10. Reproducing Kernel Hilbert Space (RKHS)
  11. Mean-zero RKHS
  12. Universal Kernels
  13. Fixed Point Analysis
  14. Debiasing with RKHS
  15. Working RKHS-Based Tangent Space
  16. ...and 33 more sections

Key Result

Proposition 3.1

Assume that $\Psi$ is pathwise differentiable at both $P^*$ and $\widehat{P}_n$, with canonical gradients $\phi_{P^*}^*\in L_0^2(P^*)$ and $\phi_{\widehat{P}_n}^*\in L_0^2(\widehat{P}_n)$. Suppose that $\Psi$ admits a second-order remainder $R_2(\cdot,\cdot)$ in a neighborhood of $\widehat{P}_n$, so Then the estimation error $\Psi(\widehat{P}_n)-\Psi(P^*)$ decomposes as

Figures (2)

  • Figure 1: Finite-sample behavior of the different ATE estimators (ULFS-KDPE, KDPE, TMLE, One-step TMLE). True asymptotic distribution is depicted in purple.
  • Figure 2: ULFS-KDPE under different stopping criteria. True asymptotic distribution is depicted in purple.

Theorems & Definitions (50)

  • Definition 1: Pathwise Differentiability
  • Definition 2: Asymptotic Linearity
  • Definition 3: Regular Estimator
  • Proposition 3.1: von Mises expansion bickel1998
  • Definition 4: Locally least favorable submodel
  • Definition 5: Universal least favorable submodel
  • Proposition 3.2: Moore--Aronszajn aronszajn50reproducing
  • Proposition 3.3: Mean-zero RKHS
  • Definition 6: Universal kernel
  • Definition 7: Contraction
  • ...and 40 more