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Abstract
This dissertation consists of four chapters regarding statistical inference on small sample size data. The first three chapters include small area estimation studies, and the last chapter addresses the high-dimensional outlier detection problem.
In the first two chapters, small area estimation models that account for heteroscedastic random effects are explored under the Bayesian framework. In particular, in Chapter 1, we study a hierarchical Bayes random regression coefficients model which accounts for the heteroscedasticity as an appropriate quadratic function of covariates. Chapter 2 considers spatially correlated random effects. Hierarchical Bayes spatial models based on four different autocorrelation structures are introduced to capture the extra variabilities caused by spatial dependence.
The third chapter of the dissertation studies measurement error models in small area estimation. In many cases, area-level models benefit from auxiliary variables that are observed with random errors such as covariates that are estimates drawn from another survey. The uncertainty in such covariates can be accounted for by fitting measurement error models. We examine and contrast two types of measurement error models with the alternative of simply ignoring the sampling errors in the covariates.
The last topic relates to high dimensional outlier detection problem. Specifically, we consider the case when the number of variables is much larger than the sample size. A randomization test called the subspace rotation test is proposed to conduct hypothesis tests for potential outliers. We justify the subspace rotation test by showing the unbiasedness of the distribution estimator and finite sample exactness. In the context of outlier detection, we also show that the power of the subspace rotation test converges to one as the dimension increases.
In the first two chapters, small area estimation models that account for heteroscedastic random effects are explored under the Bayesian framework. In particular, in Chapter 1, we study a hierarchical Bayes random regression coefficients model which accounts for the heteroscedasticity as an appropriate quadratic function of covariates. Chapter 2 considers spatially correlated random effects. Hierarchical Bayes spatial models based on four different autocorrelation structures are introduced to capture the extra variabilities caused by spatial dependence.
The third chapter of the dissertation studies measurement error models in small area estimation. In many cases, area-level models benefit from auxiliary variables that are observed with random errors such as covariates that are estimates drawn from another survey. The uncertainty in such covariates can be accounted for by fitting measurement error models. We examine and contrast two types of measurement error models with the alternative of simply ignoring the sampling errors in the covariates.
The last topic relates to high dimensional outlier detection problem. Specifically, we consider the case when the number of variables is much larger than the sample size. A randomization test called the subspace rotation test is proposed to conduct hypothesis tests for potential outliers. We justify the subspace rotation test by showing the unbiasedness of the distribution estimator and finite sample exactness. In the context of outlier detection, we also show that the power of the subspace rotation test converges to one as the dimension increases.