Strong Consistency of the SIMEX Estimator in Linear Regression with a Conditionally Poisson Covariate

This paper considers estimation for linear regression analysis with covariate measurement error arising from Poisson surrogates. We consider cases where covariates follow a conditional Poisson distribution, capturing non-Gaussian and heteroscedastic error structures. To address this, we extend the simulation extrapolation (SIMEX) algorithm to the conditional Poisson setting (POI-SIMEX), enabling robust adjustment in the absence of internal validation data. Theoretical analysis establishes strong consistency of the POI-SIMEX estimator under a linear regression framework.