Poisson regression with right-censored covariate

Right-censoring of covariates is common in count-data applications, yet standard Poisson regression typically assumes fully observed predictors. We consider a cross-sectional Poisson model with one covariate subject to random right-censoring and propose a likelihood-based estimator that incorporates both uncensored and censored observations. For censored cases, the likelihood contribution integrates over the conditional distribution beyond the censoring threshold. The resulting maximum-likelihood approach accounts for censoring-induced uncertainty without discarding data or using ad hoc substitutions. Simulation results and an empirical illustration highlight the advantages of the proposed method.