Bayesian pseudo‐empirical‐likelihood intervals for complex surveys

Summary. Bayesian methods for inference on finite population means and other parameters by using sample survey data face hurdles in all three phases of the inferential procedure: the formulation of a likelihood function, the choice of a prior distribution and the validity of posterior inferences under the design‐based frequentist framework. In the case of independent and identically distributed observations, the profile empirical likelihood function of the mean and a non‐informative prior on the mean can be used as the basis for inference on the mean and the resulting Bayesian empirical likelihood intervals are also asymptotically valid under the frequentist set‐up. For complex survey data, we show that a pseudo‐empirical‐likelihood approach can be used to construct Bayesian pseudo‐empirical‐likelihood intervals that are asymptotically valid under the design‐based set‐up. The approach proposed compares favourably with a full Bayesian analysis under simple random sampling without replacement. It is also valid under general single‐stage unequal probability sampling designs, unlike a full Bayesian analysis. Moreover, the approach is very flexible in using auxiliary population information and can accommodate two scenarios which are practically important: incorporation of known auxiliary population information for the construction of intervals by using the basic design weights; calculation of intervals by using calibration weights based on known auxiliary population means or totals.