Bayesian empirical likelihood inference with complex survey data

We propose a Bayesian empirical likelihood approach to survey data analysis on a vector of finite population parameters defined through estimating equations. Our method allows overidentified estimating equation systems and is applicable to both smooth and non‐differentiable estimating functions. Our proposed Bayesian estimator is design consistent for general sampling designs and the Bayesian credible intervals are calibrated in the sense of having asymptotically valid design‐based frequentist properties under single‐stage unequal probability sampling designs with small sampling fractions. Large sample properties of the Bayesian inference proposed are established for both non‐informative and informative priors under the design‐based framework. We also propose a Bayesian model selection procedure with complex survey data and show that it works for general sampling designs. An efficient Markov chain Monte Carlo procedure is described for the required computation of the posterior distribution for general vector parameters. Simulation studies and an application to a real survey data set are included to examine the finite sample performances of the methods proposed as well as the effect of different types of prior and different types of sampling design.