Lasso-type estimation for covariate-adjusted linear model

Lasso is popularly used for variable selection in recent years. In this paper, lasso-type penalty functions including lasso and adaptive lasso are employed in simultaneously variable selection and parameter estimation for covariate-adjusted linear model, where the predictors and response cannot be observed directly and distorted by some observable covariate through some unknown multiplicative smooth functions. Estimation procedures are proposed and some asymptotic properties are obtained under some mild conditions. It deserves noting that under appropriate conditions, the adaptive lasso estimator correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance, i.e. the adaptive lasso estimator has the oracle property in the sense of Fan and Li [6]. Simulation studies are carried out to examine its performance in finite sample situations and the Boston Housing data is analyzed for illustration.