The broken adaptive ridge estimation for the high-dimensional covariate-adjusted regression model

We consider the covariate-adjusted regression (CAR) model in which both predictors and response are not observable but distorted by a common confounding variable in multiplication way. In this paper, we employ the broken adaptive ridge (BAR) method to simultaneously select variables and estimate the coefficients for high-dimensional CAR model and obtain a CAR-BAR (CBAR) estimator after the predictors and response adjusted. We establish that CBAR estimator is consistent for variable selection and oracle for parameter estimation and possesses a grouping property for highly correlated covariates. Further, a consistent estimator of the error variance is given. Extensive simulations indicate that our CBAR method is superior to LASSO, Adaptive LASSO, SCAD and MCP methods. Particularly, a simulation scenario involving collinear predictors also demonstrates that CBAR outperforms other methods. Moreover, we show that the CBAR enjoys fast convergence rate by a straightforward simulation. In the analysis of Boston Housing data, we construct a statistic to test whether the selected variables have significant effect on the response, and give different and interesting explanations for the fitted model.