Measurement error, often treated as a missing data problem, poses major challenges in health and social science research by introducing bias and unreliable inferences. This thesis applies the Bayesian Jackknife Empirical Likelihood (BJEL) method to handle measurement error using jackknife pseudo-values based on semiparametric fractional imputation (SFI), propensity score (PS), and doubly robust (DR) estimators. Through simulation studies, BJEL was compared with traditional methods, including Jackknife Normal Approximation (JNA), Jackknife Empirical Likelihood (JEL), and Bootstrap (BS). Results showed BJEL outperformed the alternatives in terms of credible interval and average length, particularly with small sample sizes. The DR estimator showed notable robustness, maintaining reliable coverage even under model misspecification. The method was further applied to 2015–2016 NHANES data to evaluate the association between perfluoroalkyl acid (PFA) exposure and impaired kidney function.