Minimum Hellinger distance estimators for complex survey designs

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) $$L^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -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 $$\textbf{A}^{-1}\varvec{\Sigma }\textbf{A}^{-\intercal }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mi>Σ</mml:mi> </mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mo>⊺</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> (and a finite-population correction under without-replacement designs); and (iv) robustness via the influence function and $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -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.