🤖 AI Summary
This study addresses the lack of statistical inference tools for Gaussian kernel robust regression (GKRReg) by establishing, for the first time, its theoretical connection to the redescending M-estimator family through generalized M-estimation theory. Building on this foundation, the authors develop a comprehensive statistical inference framework that introduces a closed-form sandwich variance estimator—based on HC0-type heteroskedasticity-robust covariance matrices—and a paired bootstrap procedure with dynamically adjusted kernel bandwidths. Efficient computation is achieved via an iteratively reweighted least squares (IRWLS) algorithm. The resulting methodology has been implemented in the R package gkrreg, which supports diagnostic plotting, multiple γ² estimators, and benchmark datasets. Empirical evaluations on several real-world datasets demonstrate the framework’s effectiveness, substantially enhancing the practical utility and interpretability of GKRReg.
📝 Abstract
The Gaussian Kernel Robust Regression method (GKRReg) is a robust regression estimator that iteratively re-weights observations via a Gaussian kernel so that outliers and leverage points receive near-zero weight, with convergence of the estimation algorithm theoretically guaranteed. Despite a thorough study of estimation, the original work leaves open the problem of statistical inference for the regression coefficients. We fill this gap with three contributions. First, we formally establish that GKRReg belongs to the family of redescending M-estimators, providing the theoretical foundation for the inferential procedures that follow. Second, we derive a closed-form analytic sandwich variance estimator based on the theory of generalised M-estimators, corresponding to the HC0 class of heteroskedasticity-robust covariance matrices; we show that a finite-sample correction analogous to HC3 requires the weighted hat matrix of the converged IRWLS step, and identify this as a direction for future work. Third, we propose a pairs bootstrap that re-estimates the kernel width hyper-parameter gamma^2 on every replicate, capturing variability that the sandwich ignores. All procedures are implemented in the R package gkrreg, which also provides four estimators for gamma^2 and an automatic data-driven selection procedure, comprehensive diagnostic plots, and six real datasets from the robust regression literature. Applications to real data sets and comparison with traditional robust regression models highlight the potential of the GKRReg and the usability of the R package.