David Kepplinger

David Kepplinger, Anand N. Vidyashankar

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.

David Kepplinger

Heavy-tailed error distributions and predictors with anomalous values are ubiquitous in high-dimensional regression problems and can seriously jeopardize the validity of statistical analyses if not properly addressed. For more reliable estimation under these adverse conditions, we propose a new robust regularized estimator for simultaneous variable selection and coefficient estimation. This estimator, called adaptive PENSE, possesses the oracle property without prior knowledge of the scale of the residuals and without any moment conditions on the error distribution. The proposed estimator gives reliable results even under very heavy-tailed error distributions and aberrant contamination in the predictors or residuals. Importantly, even in these challenging settings variable selection by adaptive PENSE remains stable. Numerical studies on simulated and real data sets highlight superior finite-sample performance in a vast range of settings compared to other robust regularized estimators in the case of contaminated samples and competitiveness compared to classical regularized estimators in clean samples.