Semiparametric approach to regression with a covariate subject to a detection limit

We consider generalized linear regression with a covariate left-censored at a lower detection limit. Complete-case analysis, where observations with values below the limit are eliminated, yields valid estimates for regression coefficients but loses efficiency, ad hoc substitution methods are biased, and parametric maximum likelihood estimation relies on parametric models for the unobservable tail probability distribution and may suffer from model misspecification. To obtain robust and more efficient results, we propose a semiparametric likelihood-based approach using an accelerated failure time model for the covariate subject to the detection limit. A two-stage estimation procedure is developed, where the conditional distribution of this covariate given other variables is estimated prior to maximizing the likelihood function. The proposed method outperforms complete-case analysis and substitution methods in simulation studies. Technical conditions for desirable asymptotic properties are provided.