Envelope model is a nascent dimension reduction technique. We focus on extending the envelope methodology to broader applications. In the first part of this thesis we propose a common reducing subspace model that can simultaneously estimating covariance, precision matrices and their differences across multiple populations. This model leads to substantial dimension reduction and efficient parameter estimation. We explicitly quantify the efficiency gain through an asymptotic analysis. In the second part, we propose a set of new mixture models called CLEMM (Clustering with Envelope Mixture Models) that is based on the widely used Gaussian mixture model assumptions. The proposed CLEMM framework and the associated envelope-EM algorithms provides the foundations for envelope methodology in unsupervised and semi-supervised learning problems. We also illustrate the performance of these models with simulation studies and empirical applications. Also, we have extended the envelope discriminant analysis from vector data to tensor data in the third part of this thesis. Another study on copula-based models for forecasting realized volatility matrix is included, which is an important financial application of estimating covariance matrices. We consider multivariate-t, Clayton, and bivariate t, Gumbel, Clayton copulas to model and forecast one-day ahead realized volatility matrices. Empirical results show that copula based models can achieve significant performance both in terms of statistical precision and economical efficiency. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2019. / April 18, 2019. / Clustering Analysis, Dimension Reduction, EM algorithm, Envelope models, Reducing subspace, Tensor classification / Includes bibliographical references. / Xin Zhang, Professor Co-Directing Dissertation; Minjing Tao, Professor Co-Directing Dissertation; Wen Li, University Representative; Fred Huffer, Committee Member.

Many biomedical studies monitor both a longitudinal marker and a survival time on each subject under study. Modeling these two endpoints as joint responses has potential to improve the inference for both. We consider the approach of Brown and Ibrahim (2003) that proposes a Bayesian hierarchical semiparametric joint model. The model links the longitudinal and survival outcomes by incorporating the mean longitudinal trajectory as a predictor for the survival time. The usual parametric mixed effects model for the longitudinal trajectory is relaxed by using a Dirichlet process prior on the coefficients. A Cox proportional hazards model is then used for the survival time. The complicated joint likelihood increases the computational complexity. We develop a computationally efficient method by using a multivariate log-gamma distribution instead of Gaussian distribution to model the data. We use Gibbs sampling combined with Neal's algorithm (2000) and the Metropolis-Hastings method for inference. Simulation studies illustrate the procedure and compare this log-gamma joint model with the Gaussian joint models. We apply this joint modeling method to a human immunodeciency virus (HIV) data and a prostate-specific antigen (PSA) data. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2019. / April 16, 2019. / Bayesian, Gibbs Sampler, Joint model, Longitudinal, Survival / Includes bibliographical references. / Elizabeth H. Slate, Professor Co-Directing Dissertation; Jonathan R. Bradley, Professor Co-Directing Dissertation; Amy M. Wetherby, University Representative; Lifeng Lin, Committee Member.

In modern day financial risk management, modeling and forecasting stock return movements via their conditional volatilities, particularly predicting the Value at Risk (VaR), became increasingly more important for a healthy economical environment. In this dissertation, we evaluate and compare two main families of models for the conditional volatilities - GARCH and Stochastic Volatility (SV) - in terms of their VaR prediction performance of 5 major US stock indices. We calculate GARCH-type model parameters via Quasi Maximum Likelihood Estimation (QMLE) while for those of SV we employ MCMC with Ancillary Sufficient Interweaving Strategy. We use the forecast volatilities corresponding to each model to predict the VaR of the 5 indices. We test the predictive performances of the estimated models by a two-stage backtesting procedure and then compare them via the Lopez loss function. Results of this dissertation indicate that even though it is more computational demanding than GARCH-type models, SV dominates them in forecasting VaR. Since financial volatilities are moving together across assets and markets, it becomes apparent that modeling the volatilities in a multivariate framework of modeling is more appropriate. However, existing studies in the literature do not present compelling evidence for a strong preference between univariate and multivariate models. In this dissertation we also address the problem of forecasting portfolio VaR via multivariate GARCH models versus univariate GARCH models. We construct 3 portfolios with stock returns of 3 major US stock indices, 6 major banks and 6 major technical companies respectively. For each portfolio, we model the portfolio conditional covariances with GARCH, EGARCH and MGARCH-BEKK, MGARCH-DCC, and GO-GARCH models. For each estimated model, the forecast portfolio volatilities are further used to calculate (portfolio) VaR. The ability to capture the portfolio volatilities is evaluated by MAE and RMSE; the VaR prediction performance is tested through a two-stage backtesting procedure and compared in terms of the loss function. The results of our study indicate that even though MGARCH models are better in predicting the volatilities of some portfolios, GARCH models could perform as well as their multivariate (and computationally more demanding) counterparts. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2019. / April 2, 2019. / GARCH, MGARCH, SV, VaR / Includes bibliographical references. / Xufeng Niu, Professor Directing Dissertation; Giray Ökten, University Representative; Fred Huffer, Committee Member; Wei Wu, Committee Member.

Source: Dissertation Abstracts International, Volume: 40-02, Section: B, page: 0817. / Thesis (Ph.D.)--The Florida State University, 1978.

Source: Dissertation Abstracts International, Volume: 40-02, Section: B, page: 0819. / Thesis (Ph.D.)--The Florida State University, 1978.

Source: Dissertation Abstracts International, Volume: 35-02, Section: B, page: 1116. / Thesis (Ph.D.)--The Florida State University, 1973.

Source: Dissertation Abstracts International, Volume: 35-02, Section: B, page: 1117. / Thesis (Ph.D.)--The Florida State University, 1973.

Source: Dissertation Abstracts International, Volume: 36-08, Section: B, page: 4056. / Thesis (Ph.D.)--The Florida State University, 1975.

Source: Dissertation Abstracts International, Volume: 37-10, Section: B, page: 5198. / Thesis (Ph.D.)--The Florida State University, 1976.

Source: Dissertation Abstracts International, Volume: 38-11, Section: B, page: 5464. / Thesis (Ph.D.)--The Florida State University, 1977.