Some Contributions to Sampling Theory with Applications

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This project will develop methods for small area estimation. Small area estimation generally requires the use of models, either explicitly or implicitly. These model-based estimates can differ widely from the direct estimates, especially for areas with very low sample sizes. While model-based small area estimates are very useful, one potential difficulty with these estimates is that when aggregated, the overall estimate for a larger geographical area may be quite different from the corresponding direct estimate, which is usually believed to be more reliable. One way to avoid this problem is the so-called "benchmarking approach," which amounts to modifying these model-based estimates so that one gets the same aggregate estimate for the larger geographical area. This research project will develop a general two-stage Bayesian benchmarking procedure using a single model. With this approach, for example, the state per capita income estimates would be benchmarked to the national per capita income estimate and the corresponding county estimates to the benchmarked state estimates without requiring two separate models. The researcher will develop Bayesian pseudo-empirical likelihood for estimating finite population distribution functions and the corresponding population quantiles. The approach will be extended to estimation of the population distribution function in the small area context where the goal is the same but for which one needs individual estimates for local areas, often involving very small sample sizes. This makes it necessary to "borrow strength" through linking models based on auxiliary information available from censuses or other administrative records. The third component of the research involves inference under informative sampling based on copula models. The copula model to be considered in this project allows dependence in modeling the selection probabilities in terms of the observed outcomes. This is in contrast to the currently available method, which assumes independence in this modeling.The methods to be developed from this project have multiple applications and will be of value to a broad range of survey researchers. In particular, the research on two-stage benchmarks will be of relevance for many of the Federal statistical agencies. The work on empirical likelihood, with particular emphasis on estimation of small area poverty indicators, is an extremely timely topic. Finally, the research on informative sampling will be of value to researchers across many fields, including epidemiology and economics.