ADVISERS: Jie Peng / Debashis Paul
TITLE: Statistical methods for diffusion magnetic resonance imaging
ABSTRACT: We study two problems in analyzing diffusion MRI data. In the first part of this thesis, we consider tensor smoothing by extending kernel smoothing to tensor space. We conduct theoretical and numerical analysis to compare three kernel-based DTI smoothers under different geometries on the tensor space, namely the Euclidean, log-Euclidean and Affine-invariant geometries. In the second part of this thesis, we consider estimating fiber orientation distributions (FOD) based on High Angular Resolution Diffusion Imaging (HARDI). We adopt a spherical wavelet framework and utilize lasso regression to estimate the fiber directions. We also propose a data-driven procedure to determine the number of fiber directions based on an estimated FOD.
Kehui Chen (2012)
ADVISER: Hans-Georg Müller
TITLE: Modeling Conditional Distributions for Functional Data
ABSTRACT: In this talk, I will present methods for conditional distribution modeling when the predictor is in a functional space, which is an important extension of the usual functional mean regression. The first part will focus on the conditional distribution and quantile estimations for a scalar response. The study is motivated and illustrated by an application to the assessment of children’s growth patterns. The proposed method is supported by theory and is shown to perform well in simulations. In the second part, I will present results for a more complex case when responses are also functions. The method is illustrated by an application to the prediction of traffic flow, where we construct prediction regions for future speed curves.
Senke Chen (2012)
ADVISER: Jiming Jiang
TITLE: Predictive Modeling for Clustered Data with Applications
ABSTRACT: The mixed effects models are widely used for clustered data. The prediction of a random effect, or, more generally, a mixed effect is often of interest. Traditionally, the parameters are estimated based on the likelihood function, leading to the empirical best linear unbiased predictor (EBLUP) or empirical best predictor (EBP), which is essentially a hybrid of the estimation and prediction. When the underlying model is misspecified, the optimal prediction is not equivalent to the optimal estimation. In this talk, we develop a purely predictive procedure based on mean square prediction errors for LMM, GLMM and non-linear mixed effects models, and derive the best predictive estimator (BPE) of the fixed parameters, which leads to the prediction of the mixed effects, called observed best prediction (OBP). We also develop another predictive procedures based on the predictive distribution of the mixed effects for a wide range of the mixed models, and derive the weighted best predictive estimator (WBPE) of fixed parameters. The latter leads to weighted observed best prediction (WOBP).
The consistency and asymptotic distribution of the BPE and WBPE are illustrated under model misspecifications. Simulation studies and the theoretic derivations are provided to show the better performance of the BPE/WBPE over the MLE in terms of prediction criteria. We apply this approach to a workers' compensation data and a loss reserving data as applications.
Jiani Mou (2012)
ADVISER: Jiming Jiang
TITLE: Two-Stage Fence for Selecting both Fixed Covariates and Covariance Structure in Longitudinal Data
ABSTRACT: Linear Mixed models are widely used in practice, but the literature of model selection is rather sparse. Fence method (Jiang et al. 2008) is a recently developed model selection strategy and has been proposed to solve nonconventional problems. In order to make Fence methods more suitable for a wide variety of problems, such as selecting models for longitudinal autoregression data not only on regression variables but also on the variance-covariance structure, we develop a new Two-Stage Fence procedure based on certain measure of lack of fit function, an adaptive constant c and REML idea. Methodology developments are supported by simulation studies and real data analyses.
Zheng Tan (2012)
ADVISER: Rituparna Sen
TITLE: Testing Contagion in Multivariate Financial Time Series based on Residual and Recurrence Times
ABSTRACT: Financial contagion refers to the transmission of a financial shock in one entity to other interdependent entities. While the study of causes and prevention of contagion is very popular among economists, there are not many quantitative studies on how to detect (hypothesis testing) and measure (estimate) contagion. In this dissertation thesis, a new idea of Residual and Recurrence Times method of high or low values for multivariate time series is shown. With financial contagion, the distributions of residual and recurrence times are not the same, where the equality of two distributions is examined by permutation test. When compared to some methods in multivariate extreme value theory, this new method does not need the IID assumption and can handle the situation where the extremes for different components do not occur at the same time.
Ming Zhong (2012)
ADVISER: Alexander Aue
TITLE: Break Point Estimation and Variable Selection via Quantile Regressions
ABSTRACT: In both statistics and econometrics, ensuring structural stability and selecting appropriate covariates are seen as important issues. Most of the existing work in this context is done for linear regression procedures based on the conditional mean. In this talk, methodology for conditional quantiles is presented. In the first part of the talk, a new procedure is introduced that simultaneously applies structural break estimation and variable selection for a single quantile or across multiple quantiles of interest. The problem is phrased as a model selection problem in the class of piecewise quantile regressions. The best model is deﬁned in terms of minimizing a minimum description length criterion derived from an asymmetric Laplace likelihood. Its practical minimization is done with the use of genetic algorithms. If the data generating process follows indeed a piecewise quantile regression structure, it is shown that the proposed method is consistent for estimating the break points and selecting the correct variables. Numerical results from simulations and real data applications show that the new approach is competitive with and often superior to a number of existing methods.
In the second part of the talk, the methodology is extended to the ﬁtting of non-stationary time series exhibiting non-linearity, asymmetry and local persistence. This fitting is done by performing model selection in the class of piecewise stationary quantile autoregressive processes. The methodology is similar to the one developed for the quantile regression setting. Theoretical consistency results for estimating break points and autoregressive parameters are discussed. Empirical work suggests that the method performs well in ﬁnite samples.
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