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Minimum Hellinger Distance Estimators for Complex Survey Designs

David Kepplinger, Anand N. Vidyashankar

TL;DR

The paper tackles reliable inference from complex survey designs under potential outliers and high leverage induced by unequal inclusion probabilities and calibration. It introduces the Minimum Hellinger Distance Estimator (MHDE) built on a Horvitz-Thompson adjusted KDE, and proves L1-consistency of the HT KDE, consistency and exponential tail control of MHDE, and asymptotic normality with an efficient covariance structure, including a finite-population correction for without-replacement designs. Robustness is formalized via the influence function and $\alpha$-influence curves in the Hellinger topology, with simulations (Gamma and lognormal) and an NHANES application illustrating favorable efficiency-robustness trade-offs and stability against extreme responses. The estimator is computationally accessible through grid-based quadrature and is adaptable to other divergence families, offering a practical, robust tool for complex survey inference. Overall, the MHDE provides a principled, scalable approach for robust, design-aware density-based estimation in large surveys, maintaining efficiency when the model is correct and resilience under contamination or misspecification.

Abstract

Reliable inference from complex survey samples can be derailed by outliers and high-leverage observations induced by unequal inclusion probabilities and calibration. We develop a minimum Hellinger distance estimator (MHDE) for parametric superpopulation models under complex designs, including Poisson PPS and fixed-size SRS/PPS without replacement, with possibly stochastic post-stratified or calibrated weights. Using a Horvitz-Thompson-adjusted kernel density plug-in, we show: (i) $L^1$-consistency of the KDE with explicit large-deviation tail bounds driven by a variance-adaptive effective sample size; (ii) uniform exponential bounds for the Hellinger affinity that yield MHDE consistency under mild identifiability; (iii) an asymptotic Normal distribution for the MHDE with covariance $\mathbf A^{-1}\boldsymbolΣ\mathbf A^{\intercal}$ (and a finite-population correction under without-replacement designs); and (iv) robustness via the influence function and $α$-influence curves in the Hellinger topology. Simulations under Gamma and lognormal superpopulation models quantify efficiency-robustness trade-offs relative to weighted MLE under independent and high-leverage contamination. An application to NHANES 2021-2023 total water consumption shows that the MHDE remains stable despite extreme responses that markedly bias the MLE. The estimator is simple to implement via quadrature over a fixed grid and is extensible to other divergence families.

Minimum Hellinger Distance Estimators for Complex Survey Designs

TL;DR

The paper tackles reliable inference from complex survey designs under potential outliers and high leverage induced by unequal inclusion probabilities and calibration. It introduces the Minimum Hellinger Distance Estimator (MHDE) built on a Horvitz-Thompson adjusted KDE, and proves L1-consistency of the HT KDE, consistency and exponential tail control of MHDE, and asymptotic normality with an efficient covariance structure, including a finite-population correction for without-replacement designs. Robustness is formalized via the influence function and -influence curves in the Hellinger topology, with simulations (Gamma and lognormal) and an NHANES application illustrating favorable efficiency-robustness trade-offs and stability against extreme responses. The estimator is computationally accessible through grid-based quadrature and is adaptable to other divergence families, offering a practical, robust tool for complex survey inference. Overall, the MHDE provides a principled, scalable approach for robust, design-aware density-based estimation in large surveys, maintaining efficiency when the model is correct and resilience under contamination or misspecification.

Abstract

Reliable inference from complex survey samples can be derailed by outliers and high-leverage observations induced by unequal inclusion probabilities and calibration. We develop a minimum Hellinger distance estimator (MHDE) for parametric superpopulation models under complex designs, including Poisson PPS and fixed-size SRS/PPS without replacement, with possibly stochastic post-stratified or calibrated weights. Using a Horvitz-Thompson-adjusted kernel density plug-in, we show: (i) -consistency of the KDE with explicit large-deviation tail bounds driven by a variance-adaptive effective sample size; (ii) uniform exponential bounds for the Hellinger affinity that yield MHDE consistency under mild identifiability; (iii) an asymptotic Normal distribution for the MHDE with covariance (and a finite-population correction under without-replacement designs); and (iv) robustness via the influence function and -influence curves in the Hellinger topology. Simulations under Gamma and lognormal superpopulation models quantify efficiency-robustness trade-offs relative to weighted MLE under independent and high-leverage contamination. An application to NHANES 2021-2023 total water consumption shows that the MHDE remains stable despite extreme responses that markedly bias the MLE. The estimator is simple to implement via quadrature over a fixed grid and is extensible to other divergence families.
Paper Structure (28 sections, 27 theorems, 102 equations, 10 figures, 2 tables)

This paper contains 28 sections, 27 theorems, 102 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Methodology
  5. A Note on Computation
  6. Theory
  7. Consistency of the HT-adjusted KDE
  8. Consistency of the MHDE
  9. Asymptotic normality
  10. Robustness
  11. Empirical Studies
  12. Simulation study
  13. Results
  14. Application to NHANES
  15. Discussion
  16. ...and 13 more sections

Key Result

Theorem 3.1

Under Assumptions ass:kernel-1--ass:design-regularity, Moreover, there exist constants $C_1,C_2,C_3>0$, depending only on $K$ and $c_0$, such that for all $\tau\in(0,1]$, If in addition $n_{\text{eff},\gamma} h_\gamma^d / \log(1/h_\gamma)\to\infty$, then $\|\hat{f}_{\gamma}-g\|_{1}\to 0$ almost surely.

Figures (10)

  • Figure 1: Relative bias of the MHDE (blue dots) and MLE (gray triangles) in a Gamma superpopulation model using various sample designs. The sample size in each simulation is determined by $n=\alpha N$, with $\alpha \in \{10^{-3},10^{-4}\}$.
  • Figure 2: Relative RMSE of the MHDE (blue dots) and MLE (gray triangles) under a Gamma superpopulation model using various sample designs. The sample size in each simulation is determine by $n=\alpha N$, with $\alpha \in \{10^{-3},10^{-4}\}$.
  • Figure 3: Influence functions (top) and alpha curves (bottom) for the MHDE and MLE of the scale ($\circ$) and shape ($\triangle$) parameters in the Gamma model. The contamination proportion in the influence function at the top is set to $\varepsilon=0.1$. The horizontal axis shows the location of the point-mass contamination in terms of the quantile of the true superpopulation model, i.e., $z=G^{-1}(p)$. For the alpha curve the point-mass contamination is located at $z=G^{-1}(1-10^{-7}) \approx 669\,193$.
  • Figure 4: MHDE and MLE estimates for two different parametric models to describe the total daily water consumption in the NHANES survey. The minimum Hellinger distance (MHD) is achieved by the MHDE shown here.
  • Figure 5: Relative bias of the MHDE (blue dots) and MLE (gray triangles) in a Gamma superpopulation model using calibrated weights and various sampling designs. The sample size in each simulation is determined by $n=\alpha N$, with $\alpha \in \{10^{-3},10^{-4}\}$.
  • ...and 5 more figures

Theorems & Definitions (60)

  • Theorem 3.1: Large-deviation-based $L_1$-consistency of HT-adjusted KDE
  • Proposition 3.2: Direct large-deviation bounds for the design term
  • Remark 3.3: Rates under smoothness
  • Corollary 3.4: Simple random sampling
  • Proposition 3.5: Exponential tail bounds for uniform MHDE deviation
  • Theorem 3.6: Consistency of MHDE with HT plug-in
  • proof
  • Remark 3.7
  • Remark 3.8
  • Remark 3.9
  • ...and 50 more