Bayes Estimation and Prediction under Informative Sampling Design

In this research we will deal with the problem of Bayes estimation of the parameter that characterise the superpopulation model, and Bayes prediction of finite population total, from a sample survey data selected from a finite population using informative probability sampling design, that is, the sample first order inclusion probabilities depend on the values of the model outcome variable (or the model outcome variable is correlated with design variables not included in the model). In order to achieve this we will first define the sample predictive distribution and the sample posterior distribution and then we use the sample posterior likelihood function to obtain the sample Bayes estimate of the superpopulation model parameters, and Bayes predictors of finite population total. These new predictors take into account informative sampling design. Thus, provides new justification for the broad use of best linear unbiased predictors (model-based school) in predicting finite population parameters in case of not accounting of complex sampling design. Furthermore, we show that the behaviours of the present estimators and predictors depends on the informativeness parameters. Also the use of the Bayes estimators and predictors that ignore the informative sampling design yields biased Bayes estimators and predictors. One of the most important feature of this paper is, specifying prior distribution for the parameters of the sample distribution makes life easier than the population parameters