High dimensional covariance matrix estimation is considered in the context of empirical asset pricing. In order to see the effects of covariance matrix estimation on asset pricing, parameter estimation, model specification test, and misspecification problems are explored. Along with existing techniques, which is not yet tested in applications, diagonal variance matrix is simulated to evaluate the performances in these problems. We found that modified Stein type estimator outperforms all the other methods in all three cases. In addition, it turned out that heuristic method of diagonal variance matrix works far better than existing methods in Hansen-Jagannathan distance test. High dimensional covariance matrix as a transformation matrix in generalized least squares is also studied. Since the feasible generalized least squares estimator requires ex ante knowledge of the covariance structure, it is not applicable in general cases. We propose fully banding strategy for the new estimation technique. First we look into the sparsity of covariance matrix and the performances of GLS. Then we move onto the discussion of diagonals of covariance matrix and column summation of inverse of covariance matrix to see the effects on GLS estimation. In addition, factor analysis is employed to model the covariance matrix and it turned out that communality truly matters in efficiency of GLS estimation.

This century is surely the century of data (Donoho, 2000). Data analysis has been an emerging activity over the last few decades. High dimensional data is in particular more and more pervasive with the advance of massive data collection system, such as microarrays, satellite imagery, and financial data. However, analysis of high dimensional data is of challenge with the so called curse of dimensionality (Bellman 1961). This research dissertation presents several methodologies in the application of high dimensional data analysis. The first part discusses a joint analysis of multiple microarray gene expressions. Microarray analysis dates back to Golub et al. (1999). It draws much attention after that. One common goal of microarray analysis is to determine which genes are differentially expressed. These genes behave significantly differently between groups of individuals. However, in microarray analysis, there are thousands of genes but few arrays (samples, individuals) and thus relatively low reproducibility remains. It is natural to consider joint analyses that could combine microarrays from different experiments effectively in order to achieve improved accuracy. In particular, we present a model-based approach for better identification of differentially expressed genes by incorporating data from different studies. The model can accommodate in a seamless fashion a wide range of studies including those performed at different platforms, and/or under different but overlapping biological conditions. Model-based inferences can be done in an empirical Bayes fashion. Because of the information sharing among studies, the joint analysis dramatically improves inferences based on individual analysis. Simulation studies and real data examples are presented to demonstrate the effectiveness of the proposed approach under a variety of complications that often arise in practice. The second part is about covariance matrix estimation in high dimensional data. First, we propose a penalised likelihood estimator for high dimensional t-distribution. The student t-distribution is of increasing interest in mathematical finance, education and many other applications. However, the application in t-distribution is limited by the difficulty in the parameter estimation of the covariance matrix for high dimensional data. We show that by imposing LASSO penalty on the Cholesky factors of the covariance matrix, EM algorithm can efficiently compute the estimator and it performs much better than other popular estimators. Secondly, we propose an estimator for high dimensional Gaussian mixture models. Finite Gaussian mixture models are widely used in statistics thanks to its great flexibility. However, parameter estimation for Gaussian mixture models with high dimensionality can be rather challenging because of the huge number of parameters that need to be estimated. For such purposes, we propose a penalized likelihood estimator to specifically address such difficulties. The LASSO penalty we impose on the inverse covariance matrices encourages sparsity on its entries and therefore helps reducing the dimensionality of the problem. We show that the proposed estimator can be efficiently computed via an Expectation-Maximization algorithm. To illustrate the practical merits of the proposed method, we consider its application in model-based clustering and mixture discriminant analysis. Numerical experiments with both simulated and real data show that the new method is a valuable tool in handling high dimensional data. Finally, we present structured estimators for high dimensional Gaussian mixture models. The graphical representation of every cluster in Gaussian mixture models may have the same or similar structure, which is an important feature in many applications, such as image processing, speech recognition and gene network analysis. Failure to consider the sharing structure would deteriorate the estimation accuracy. To address such issues, we propose two structured estimators, hierarchical Lasso estimator and group Lasso estimator. An EM algorithm can be applied to conveniently solve the estimation problem. We show that when clusters share similar structures, the proposed estimator perform much better than the separate Lasso estimator.

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Thesis (M.A.)--Carleton University, 1997. / Also available in electronic format on the Internet.

Thesis (M. A.)--University of Notre Dame, 2005. / Thesis directed by Ke-Hai Yuan for the Department of Psychology. "October 2005." Includes bibliographical references (leaves 60-63).

Thesis (Ph.D. (Econometrics))-University of Pretoria, 2005. / Abstract in English. Includes bibliographical references. Available on the Internet via the World Wide Web.

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (May 1, 2007) Vita. Includes bibliographical references.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-10T16:15:35Z No. of bitstreams: 1 LYA RAQUEL OLIVEIRA DE SOUSA - DISSERTAÇÃO PPGMAT 2006..pdf: 539595 bytes, checksum: 9c8bde69fa077d93bd6a4ec01ebd5171 (MD5) / Made available in DSpace on 2018-07-10T16:15:35Z (GMT). No. of bitstreams: 1 LYA RAQUEL OLIVEIRA DE SOUSA - DISSERTAÇÃO PPGMAT 2006..pdf: 539595 bytes, checksum: 9c8bde69fa077d93bd6a4ec01ebd5171 (MD5) Previous issue date: 2006-03 / Capes / Neste trabalho apresentamos como a inferência Bayesiana e a relação entre região H.P.D. e os testes de hipóteses, obtida através de algumas propriedades da distribuição t-Studente multivariada, podem ser aplicados no estudo de dados de um Modelo de Covariância Linear com e sem erros nas variáveis. Para o nosso trabalho, consideramos um experimento planejado em k tratamentos, sendo cada um deles repetido em ni unidades experimentais, i=1,2,...,k. / In this work we present as the Bayesian inference and the relation between region H.P.D. and the tests of hypothesis, gotten through some properties of the distribution t-Student multivaried, they can to be applied in the study of data of a Model of Linear Covariance with and without errors in the variables. For our work, we consider an experiment planned in k treatments, being each one of them repeated in ni experimental units, i=1,2,...,k.

The focus of this thesis is on the investigation of the geodesics of ruled surfaces.