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Abstract:
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We consider the problem of integrating a small probability sample (ps) and a non-probability sample (nps). By definition, for the nps, there are no survey weights, but for the ps, there are survey weights. The key issue is that the nps, although much larger than the ps, can lead to a biased estimator of a finite population quantity but with much smaller variance. We begin with a relatively simple problem in which the population is assumed to be homogeneous and there are no common units in the ps and the nps. We assume that there are covariates and responses for everyone in the two samples, and there are no covariates available for nonsampled units. We use the nps to construct a prior for the ps, particularly in small area estimation. We also introduce partial discounting to avoid a dominance of the prior. Inverse probability weighted estimators are used to do Bayesian predictive inference of the finite population mean. We show how to extend our procedure to cover estimation for small areas, where auxiliary information is much needed, and the nps is used as the prior with partial discounting. In our illustrative example on body mass index and our simulation study, we compare our procedures with inference from the ps only estimate. Our procedure provides improved estimates over the ps only.
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