| Chapter 1 is concerned with confidence interval construction for the mean of a long-range dependent time series. It is well known that the moving block bootstrap method produces an inconsistent estimator of the distribution of the normalized sample mean when its limiting distribution is not normal. The subsampling method of Hall, Lahiri and Jing (1998) produces a consistent estimator but involves consistent estimation of the variance of the normalized sample mean using one seemingly arbitrary tuning parameter. By adopting a self-normalization idea, we modify the subsampling procedure of Hall et al.(1998) and the resulting procedure does not require consistent variance estimation. The modified subsampling procedure only involves the choice of the subsampling widow width, which can be addressed by using some existing data driven selection methods. Simulations studies are conducted to compare the finite sample performances. The behavior of cluster analysis under different distance measures is explored in Chapter 2, using some of the most common models in educational testing for data generation. Theoretical results on clustering accuracy are given for distance measures used in minimum diameter partitioning and hierarchical agglomerative cluster analysis with complete linkage for data from unidimensional item response models, restricted latent class models for cognitive diagnosis, and the linear factor analysis model. An aim is to identify distance measures that work well for a variety of models, explore how much knowledge of the underlying model is needed to construct a distance measure that leads to a consistent solution, and provide theoretical justifications for using them. Clustering consistency is defined on the space of the latent trait, and consistency and inconsistency results are given for competing distance measures.;We study response times in computerized adaptive testing in Chapter 3. We propose a semi-parametric model for response times that arises in educational assessment data. Algorithms for item selection that use the response time information are proposed and studied for their efficiency and how well they distribute item exposure. |
