An iterative matrix uncertainty selector for high-dimensional generalized linear models with measurement errors.

Measurement error is a prevalent issue in high-dimensional generalized linear regression that existing regularization techniques may inadequately address. Most require estimating error distributions, which can be computationally prohibitive or unrealistic. We introduce an error distribution-free approach for variable selection called the Iterative Matrix Uncertainty Selector (IMUS). IMUS employs the matrix uncertainty selector framework for linear models, which is known for its selection consistency properties. It features an efficient iterative algorithm easily implemented for any generalized linear model within the exponential family. Empirically, we demonstrate that IMUS performs well in simulations and on three microarray gene expression datasets, achieving effective covariate selection with smoother convergence and clearer elbow criteria compared to other error distribution free methods. Notably, simulation studies in logistic and Poisson regression showed that IMUS exhibited smoother convergence and clearer elbow criteria, performing comparably to the Generalized Matrix Uncertainty Selector (GMUS) and Generalized Matrix Uncertainty Lasso (GMUL) in covariate selection. In many scenarios, IMUS had smaller estimation errors than GMUL and GMUS, measured by both the 1- and 2-norms. In applications to three microarray datasets with noisy measurements, GMUS faced convergence issues, while GMUL converged but lacked well-defined elbows for two datasets. In contrast, IMUS converged with well-defined elbows for all datasets, providing a potentially effective solution for high dimensional regression problems involving measurement errors.