Supplementing a Non-probability Sample With a Probability Sample to Predict the Finite Population Mean

We show how to analyze a non-probability sample (nps) with limited information from a small probability sample (ps). The most practical case is when the nps has auxiliary variables and study variable but no survey weights and the ps has  known weights, auxiliary variables, but no study variable. Two samples are taken from the same population and the variables are common to both the nps and the ps. A large non-probability sample can reduce the cost but will give biased estimator with small variance, the small probability sample can provide supplemental information. Following this, we apply these weights to fit a mixture model, enhancing the robustness of the results and enabling the estimation of the finite population mean. Additionally, we present a method to enhance the efficiency of the Gibbs sampler.