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Bootstrap Consistency for Empirical Likelihood in Density Ratio Models

Weiwei Zhuang, Weiqi Yang, Jiahua Chen

TL;DR

The paper addresses statistical inference under the density ratio model (DRM) using empirical likelihood, focusing on the bootstrap's validity for both DRM parameters and distribution-function functionals. It develops a rigorous theory showing that bootstrap estimators of parameters and distribution functions share the same limiting laws as their population counterparts, extending pointwise convergence to weak convergence of processes and enabling inferential tools for quantiles and dominance indices. The results include detailed asymptotic normality for the parameter estimators, weak convergence of distribution-function processes, and Hadamard-differentiable functional delta-method arguments for quantiles, all with bootstrap analogues. Simulations and a real-data example demonstrate accurate finite-sample performance and practical applicability for poverty rates, median incomes, and distributional dominance across populations.

Abstract

We establish the validity of bootstrap methods for empirical likelihood (EL) inference under the density ratio model (DRM). In particular, we prove that the bootstrap maximum EL estimators share the same limiting distribution as their population counterparts, both at the parameter level and for distribution functionals. Our results extend existing pointwise convergence theory to weak convergence of processes, which in turn justifies bootstrap inference for quantiles and dominance indices within the DRM framework. These theoretical guarantees close an important gap in the literature, providing rigorous foundations for resampling-based confidence intervals and hypothesis tests. Simulation studies further demonstrate the accuracy and practical value of the proposed approach.

Bootstrap Consistency for Empirical Likelihood in Density Ratio Models

TL;DR

The paper addresses statistical inference under the density ratio model (DRM) using empirical likelihood, focusing on the bootstrap's validity for both DRM parameters and distribution-function functionals. It develops a rigorous theory showing that bootstrap estimators of parameters and distribution functions share the same limiting laws as their population counterparts, extending pointwise convergence to weak convergence of processes and enabling inferential tools for quantiles and dominance indices. The results include detailed asymptotic normality for the parameter estimators, weak convergence of distribution-function processes, and Hadamard-differentiable functional delta-method arguments for quantiles, all with bootstrap analogues. Simulations and a real-data example demonstrate accurate finite-sample performance and practical applicability for poverty rates, median incomes, and distributional dominance across populations.

Abstract

We establish the validity of bootstrap methods for empirical likelihood (EL) inference under the density ratio model (DRM). In particular, we prove that the bootstrap maximum EL estimators share the same limiting distribution as their population counterparts, both at the parameter level and for distribution functionals. Our results extend existing pointwise convergence theory to weak convergence of processes, which in turn justifies bootstrap inference for quantiles and dominance indices within the DRM framework. These theoretical guarantees close an important gap in the literature, providing rigorous foundations for resampling-based confidence intervals and hypothesis tests. Simulation studies further demonstrate the accuracy and practical value of the proposed approach.
Paper Structure (11 sections, 10 theorems, 111 equations, 3 figures, 11 tables)

This paper contains 11 sections, 10 theorems, 111 equations, 3 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Main Results
  4. Bootstrap Estimators of Parameters
  5. Bootstrap Estimators of Distribution Functions
  6. Applications
  7. Quantile Functions
  8. Dominance Index
  9. Simulation
  10. Real-data analysis
  11. Conclusions

Key Result

Lemma 1

Under Assumption as1, the MELE is asymptotically normal: as $n\to\infty$,

Figures (3)

  • Figure 1: Empirical distribution functions for the six provinces
  • Figure 2: Income histograms overlaid with DRM-implied density estimates.
  • Figure 3: DRM estimates and bootstrap $95\%$ CIs for various indicators.

Theorems & Definitions (22)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • proof
  • ...and 12 more