1 Introduction
Jeremy Strief and Glen Meeden
Weights play an important role in the design based approach to survey sampling. In theory the weight assigned to an observed unit in a sample is the reciprocal of its selection probability and is interpreted as the number of units in the population which it represents. In practice, after a sample has been observed, the weights are often adjusted to make the sample better represent the population. These adjustments can be made to take into account population information not included in the design and for observations missing from the sample. Although such modifications of the design based weights are undoubtedly useful in some cases their ultimate theoretical justification is not so clear. Part of the confusion, we believe, comes from arguing unconditionally before the sample is taken, e.g., the Horvitz-Thompson estimator is unbiased averaged over all possible samples, and then conditionally after the sample is in hand, by adjusting the designed based weights of the observed units in the sample. In particular, an overemphasis on the sampling design at the second or conditional stage can needlessly complicated matters. After the sample has been observed, we believe a better approach is to formally ignore the sampling design but use all the available information, including that embedded in the design, to find a sensible set of weights. In this way of thinking a weight assigned to a unit can still be interpreted as the number of units in the population that it represents but it is no longer derived as an adjustment of its selection probability. How can this be done?
In the Bayesian approach information about the population is incorporated into a prior distribution. In theory, the prior can then be used to purposely select an optimal sample; however this is almost never done. After the sample is observed inferences are based on the posterior distribution of the unobserved units in the population given the values of the observed units in the sample. In most situations the posterior does not depend on how the sample was selected and hence the design plays no role at the inference stage. Bayes methods have been little used in practice because it is difficult to find prior distributions which reflect the common kinds of available prior information.
Many of the standard estimators can be given a stepwise Bayesian interpretation (Ghosh and Meeden 1997). In this approach, given any sample, inference is still based on a posterior distribution but the collection (for all possible samples) of the posteriors does not arise from a single prior but from a whole family of prior distributions. In the situation where one believes that the observed units are roughly exchangeable with the unobserved units the appropriate stepwise Bayes posterior distribution is the Polya posterior.
When prior information about population means and quantiles of auxiliary variables is available Lazar, Meeden and Nelson (2008) argued that the constrained Polya posterior, a generalization of the Polya posterior, is a sensible way to incorporate such prior information. Here we will show how the constrained Polya posterior can be used to define weights for the units in the sample. Although the resulting weights depend on the auxiliary variables they do not make explicit use of the sampling design.
In Section 2 we review the Polya posterior and in Section 3, the constrained Polya posterior. The two main ideas of the paper are given in the next two sections. In Section 4 we show how the constrained Polya posterior can be used to attached a weight to each unit in the sample and in such a way that these weights do not depend directly on the sampling design. In Section 5 we introduce the weighted Dirichlet posterior as a companion to the constrained Polya posterior. It allows one to use the weights defined by the constrained Polya posterior to make inferences about population parameters through straight forward simulation. In Section 6 we compare the constrained Polya posterior weights to those used in the Horvitz-Thompson estimator. In Section 7 we consider several examples to see how the resulting weights preform in practice and show how the weighted Dirichlet posterior can be use to get an estimate of variance for an estimator without extensive computing. Section 8 contains some concluding remarks.
At first reading it will seem to some that the methods proposed here are very Bayesian because all of our inferences are based on "posterior� distributions. But as mentioned above, technically, our "posterior� distributions are not Bayes but stepwise Bayes. This means that operationally one can think of our posterior as being constructed after the sample has been observed. These constructed "posteriors� do not depend on subjective prior information or the sampling design but just use the observed sample values and objective and public information about the auxiliary variables. As we shall see this allows one to construct estimators of population parameters which are approximately unbiased under a variety of designs and have good frequentist properties. There are two important limitations of our work however. The first is that it only is applicable to single stage designs and the second is that it cannot correct for selection bias.
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