Bayesian Predictive Inference Under Informative Sampling and Transformation Public
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We have considered the problem in which a biased sample is selected from a finite population, and this finite population itself is a random sample from an infinitely large population, called the superpopulation. The parameters of the superpopulation and the finite population are of interest. There is some information about the selection mechanism in that the selection probabilities are linearly related to the measurements. This is typical of establishment surveys where the selection probabilities are taken to be proportional to the previous year's characteristics. When all the selection probabilities are known, as in our problem, inference about the finite population can be made, but inference about the distribution is not so clear. For continuous measurements, one might assume that the the values are normally distributed, but as a practical issue normality can be tenuous. In such a situation a transformation to normality may be useful, but this transformation will destroy the linearity between the selection probabilities and the values. The purpose of this work is to address this issue. In this light we have constructed two models, an ignorable selection model and a nonignorable selection model. We use the Gibbs sampler and the sample importance re-sampling algorithm to fit the nonignorable selection model. We have emphasized estimation of the finite population parameters, although within this framework other quantities can be estimated easily. We have found that our nonignorable selection model can correct the bias due to unequal selection probabilities, and it provides improved precision over the estimates from the ignorable selection model. In addition, we have described the case in which all the selection probabilities are unknown. This is useful because many agencies (e.g., government) tend to hide these selection probabilities when public-used data are constructed. Also, we have given an extensive theoretical discussion on Poisson sampling, an underlying sampling scheme in our models especially useful in the case in which the selection probabilities are unknown.
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