Sparse Minimum Distance Kernel Estimation for High‐Dimensional Linear Models

ABSTRACT In recent years, the rapid advancement of data collection and processing technologies has led to a growing influx of high‐dimensional data into various scientific fields. However, this surge in data is often accompanied by the frequent occurrence of outliers, particularly in domains such as genomic and image data. If these outliers are not appropriately addressed, they can result in biased or invalid models that may mislead subsequent statistical inference based on the data. Building upon the minimum distance and kernel density estimator, this paper thus proposes a new method named minimum distance kernel estimation (MDKE) to overcome the problems caused by the outliers in data. We construct the objective function of MDKE by employing the kernel technique. The advantage of MDKE over existing approaches is that it is robust to both outliers and high‐leverage points and does not rely on a prespecified parametric form for the error distribution. Furthermore, the sparse MDKE (SMDKE) is proposed for high‐dimensional data by inducing the regularizations such as LASSO, SCAD and MCP. The extensive simulation studies are conducted, demonstrating that MDKE exhibits superior robustness than the existing methods for data with outliers in both the response and covariates. Analyses of real‐world datasets involving plasma beta‐carotene levels and microarray gene expression further demonstrate the superior robustness and variable selection performance of the proposed SMDKE method compared to existing alternatives.