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This thesis deals with estimation and inference in two semiparametric problems: a two-component mixture model and a single index regression model. For the two-component mixture model, we assume that the distribution of one component is known and develop methods for estimating the mixing proportion and the unknown distribution using ideas from shape restricted function estimation. We establish the consistency of our estimators. We find the rate of convergence and the asymptotic limit of our estimator for the mixing proportion. Furthermore, we develop a completely automated distribution-free honest finite sample lower confidence bound for the mixing proportion. We compare the proposed estimators, which are easily implementable, with some of the existing procedures through simulation studies and analyse two data sets, one arising from an application in astronomy and the other from a microarray experiment. For the single index model, we consider estimation of the unknown link function and the finite dimensional index parameter. We study the problem when the true link function is assumed to be: (1) smooth or (2) convex. When the link function is just assumed to be smooth, in contrast to standard kernel based methods, we use smoothing splines to estimate the link function. We prove the consistency and find the rates of convergence of the proposed estimators. We establish root-n-rate of convergence and the semiparametric efficiency of the parametric component under mild assumptions. When the link function is assumed to be convex, we propose a shape constrained penalized least squares estimator and a Lipschitz constrained least squares estimator for the unknown quantities. We prove the consistency and find the rates of convergence for both estimators. For the shape constrained penalized least squares estimator, we establish root-n-rate of convergence and the semiparametric efficiency of the parametric component under mild assumptions and conjecture that the parametric component of the Lipschitz constrained least squares estimator is semiparametrically efficient. We develop the R package "simest'' that can be used (to compute the proposed estimators) even for moderately large dimensions.

In general model selection so far considered in literature, the parameter estimation loss and the prediction loss from the model selected are considered to be the same. In this thesis, the methods of parameter estimation may vary with different estimation loss, and the model selection may be based on different prediction loss. Under some regularized conditions, a model selection criterion, called generalized risk criterion (GRC), is proposed with a closed form. For multivariate linear regression model, and Cox regression model for ranking data, our studies that this criterion is an extension of the model selection criterion AIC. We also demonstrate that GRC performs better than AIC in a practical semi-parametric regression problem involving investments on horse racing. / Keywords: Variable selection; Model selection criterion; AIC; GRC; Loss function; Risk function; Multinomial Choice Model; Cox model for ranking data. / Searching for the true model based on the limited data is usually an impossible task. More and more attention in research has been focused on how to find an optimal model based on some special objective, such as focused information criterion (FIC, Hjort and Claeskens, 2003 [15]), Subspace Information criterion (Sugiyama and Ogawa, 2001 [43]) in statistical learning, etc. These ideas also motivate us to find an optimal subset of variables based on some objective. Different objectives may result in different choices of subset of variables. / Variable selection, an important aspect of model selection, is applied widely in real practices to explore the latent relationship between the random phenomena and various factors. Many model selection criteria, such as Mallow's Cp (Mallows, 1964 [28]). PRESS (Allen, 1971 [3]). AIC (Akaike, 1973 [2]), are proposed for seeking the optimal subset of the variables. Most of them try to find a criterion based on the observed data such that the selected models perform well both for fitting and for prediction. / Zuo, Guo Xin. / "July 2007." / Adviser: Ming Gao Gu. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0402. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 71-75) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

by Ka-Kit Tse. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references. / Acknowledgments --- p.i / List of tables --- p.v / Chapter Chapter 1: --- Overall Review --- p.1 / Chapter Chapter 2: --- Essay one - A Mathematical Programming Approach to Clusterwise Regression Model and its Extensions / Chapter 2.0. --- Abstract --- p.5 / Chapter 2.1. --- Introduction --- p.6 / Chapter 2.2. --- A Mathematical Programming Formulation of the Clusterwise Regression Model --- p.10 / Chapter 2.2.1. --- The Generalized Clusterwise Regression Model --- p.10 / Chapter 2.2.2. --- "Clusterwise Regression Model (Spath, 1979)" --- p.14 / Chapter 2.2.3. --- A Nonparametric Clusterwise Regression Model --- p.15 / Chapter 2.2.4. --- A Mixture Approach to Clusterwise Regression Model --- p.16 / Chapter 2.2.5. --- An Illustrative Application --- p.19 / Chapter 2.3. --- Mathematical Programming Formulation of the Clusterwise Discriminant Analysis --- p.21 / Chapter 2.4. --- Conclusion --- p.25 / Chapter 2.5. --- Appendix --- p.28 / Chapter 2.6. --- References --- p.32 / Chapter 2.7. --- Tables --- p.35 / Chapter Chapter 3: --- Essay two - A Mathematical Programming Approach to Clusterwise Rank Order Logit Model / Chapter 3.0. --- Abstract --- p.40 / Chapter 3.1. --- Introduction --- p.41 / Chapter 3.2. --- Clusterwise Rank Order Logit Model --- p.42 / Chapter 3.3. --- Numerical Illustration --- p.46 / Chapter 3.4. --- Conclustion --- p.48 / Chapter 3.5. --- References --- p.50 / Chapter 3.6. --- Tables --- p.52 / Chapter Chapter 4: --- Essay three - A Mathematical Programming Approach to Metric Unidimensional Scaling / Chapter 4.0. --- Abstract --- p.53 / Chapter 4.1. --- Introduction --- p.54 / Chapter 4.2. --- Nonlinear Programming Formulation --- p.56 / Chapter 4.3. --- Numerical Examples --- p.60 / Chapter 4.4. --- Possible Extensions --- p.61 / Chapter 4.5. --- Conclusion and Extensions --- p.63 / Chapter 4.6. --- References --- p.64 / Chapter 4.7. --- Tables --- p.66 / Chapter Chapter 5: --- Research Project in Progress / Chapter 5.1. --- Project 1 -- An Integrated Approach to Taste Test Experiment Within the Prospect Theory Framework --- p.68 / Chapter 5.1.1. --- Experiment Procedure --- p.68 / Chapter 5.1.2. --- Experimental Result --- p.72 / Chapter 5.2. --- Project 2 -- An Integrated Approach to Multi- Dimensional Scaling Problem --- p.75 / Chapter 5.2.1. --- Introduction --- p.75 / Chapter 5.2.2. --- Experiment Procedure --- p.76 / Chapter 5.2.3. --- Questionnaire --- p.78 / Chapter 5.2.4. --- Experimental Result --- p.78

Lui Ying Kin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 63-65). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Introduction to bootstrap and efficiency bootstrap simulation --- p.5 / Chapter 2.1 --- Background of bootstrap --- p.5 / Chapter 2.2 --- Basic idea of bootstrap --- p.7 / Chapter 2.3 --- Variance reduction methods --- p.10 / Chapter 2.3.1 --- Control variates --- p.10 / Chapter 2.3.2 --- Common random numbers --- p.12 / Chapter 2.3.3 --- Antithetic variates --- p.14 / Chapter 2.3.4 --- Importance Sampling --- p.15 / Chapter 2.4 --- Efficient bootstrap simulation --- p.17 / Chapter 2.4.1 --- Linear approximation --- p.18 / Chapter 2.4.2 --- Centring method --- p.19 / Chapter 2.4.3 --- Balanced resampling --- p.20 / Chapter 2.4.4 --- Antithetic resampling --- p.21 / Chapter 3 --- Methodology --- p.22 / Chapter 3.1 --- Introduction --- p.22 / Chapter 3.2 --- Cluster analysis --- p.24 / Chapter 3.3 --- Regression estimator and mixture experiment --- p.25 / Chapter 3.4 --- Estimate of standard error and bias --- p.30 / Chapter 4 --- Simulation study --- p.45 / Chapter 4.1 --- Introduction --- p.45 / Chapter 4.2 --- Ratio estimation --- p.46 / Chapter 4.3 --- Time series problem --- p.50 / Chapter 4.4 --- Regression problem --- p.54 / Chapter 5 --- Conclusion and discussion --- p.60 / Reference --- p.63

Yang, Haiqin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 98-109). / Abstracts in English and Chinese. / Abstract --- p.ii / Acknowledgement --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Time Series Prediction and Its Problems --- p.1 / Chapter 1.2 --- Major Contributions --- p.2 / Chapter 1.3 --- Thesis Organization --- p.3 / Chapter 1.4 --- Notation --- p.4 / Chapter 2 --- Literature Review --- p.5 / Chapter 2.1 --- Framework --- p.6 / Chapter 2.1.1 --- Data Processing --- p.8 / Chapter 2.1.2 --- Model Building --- p.10 / Chapter 2.1.3 --- Forecasting Procedure --- p.12 / Chapter 2.2 --- Model Descriptions --- p.13 / Chapter 2.2.1 --- Linear Models --- p.15 / Chapter 2.2.2 --- Non-linear Models --- p.17 / Chapter 2.2.3 --- ARMA Models --- p.21 / Chapter 2.2.4 --- Support Vector Machines --- p.23 / Chapter 3 --- Support Vector Regression --- p.27 / Chapter 3.1 --- Regression Problem --- p.27 / Chapter 3.2 --- Loss Function --- p.29 / Chapter 3.3 --- Kernel Function --- p.34 / Chapter 3.4 --- Relation to Other Models --- p.36 / Chapter 3.4.1 --- Relation to Support Vector Classification --- p.36 / Chapter 3.4.2 --- Relation to Ridge Regression --- p.38 / Chapter 3.4.3 --- Relation to Radial Basis Function --- p.40 / Chapter 3.5 --- Implemented Algorithms --- p.40 / Chapter 4 --- Margins in Support Vector Regression --- p.46 / Chapter 4.1 --- Problem --- p.47 / Chapter 4.2 --- General ε-insensitive Loss Function --- p.48 / Chapter 4.3 --- Accuracy Metrics and Risk Measures --- p.52 / Chapter 5 --- Margin Variation --- p.55 / Chapter 5.1 --- Non-fixed Margin Cases --- p.55 / Chapter 5.1.1 --- Momentum --- p.55 / Chapter 5.1.2 --- GARCH --- p.57 / Chapter 5.2 --- Experiments --- p.58 / Chapter 5.2.1 --- Momentum --- p.58 / Chapter 5.2.2 --- GARCH --- p.65 / Chapter 5.3 --- Discussions --- p.72 / Chapter 6 --- Relation between Downside Risk and Asymmetrical Margin Settings --- p.77 / Chapter 6.1 --- Mathematical Derivation --- p.77 / Chapter 6.2 --- Algorithm --- p.81 / Chapter 6.3 --- Experiments --- p.83 / Chapter 6.4 --- Discussions --- p.86 / Chapter 7 --- Conclusion --- p.92 / Chapter A --- Basic Results for Solving SVR --- p.94 / Chapter A.1 --- Dual Theory --- p.94 / Chapter A.2 --- Standard Method to Solve SVR --- p.96 / Bibliography --- p.98

The overall theme of this thesis focuses on methods for functional regression and nonlinear mixed-effects models with applications to PET data. The first part considers the problem of variable selection in regression models with functional responses and scalar predictors. We pose the function-on-scalar model as a multivariate regression problem and use group-MCP for variable selection. We account for residual covariance by "pre-whitening" using an estimate of the covariance matrix, and establish theoretical properties for the resulting estimator. We further develop an iterative algorithm that alternately updates the spline coefficients and covariance. Our method is illustrated by the application to two-dimensional planar reaching motions in a study of the effects of stroke severity on motor control. The second part introduces a functional data analytic approach for the estimation of the IRF, which is necessary for describing the binding behavior of the radiotracer. Virtually all existing methods have three common aspects: summarizing the entire IRF with a single scalar measure; modeling each subject separately; and the imposition of parametric restrictions on the IRF. In contrast, we propose a functional data analytic approach that regards each subject's IRF as the basic analysis unit, models multiple subjects simultaneously, and estimates the IRF nonparametrically. We pose our model as a linear mixed effect model in which shrinkage and roughness penalties are incorporated to enforce identifiability and smoothness of the estimated curves, respectively, while monotonicity and non-negativity constraints impose biological information on estimates. We illustrate this approach by applying it to clinical PET data. The third part discusses a nonlinear mixed-effects modeling approach for PET data analysis under the assumption of a compartment model. The traditional NLS estimators of the population parameters are applied in a two-stage analysis, which brings instability issue and neglects the variation in rate parameters. In contrast, we propose to estimate the rate parameters by fitting nonlinear mixed-effects (NLME) models, in which all the subjects are modeled simultaneously by allowing rate parameters to have random effects and population parameters can be estimated directly from the joint model. Simulations are conducted to compare the power of detecting group effect in both rate parameters and summarized measures of tests based on both NLS and NLME models. We apply our NLME approach to clinical PET data to illustrate the model building procedure.

This thesis uses methods from functional data analysis (FDA) to solve problems from three scientific areas of study. While the areas of application are quite distinct, the common thread of functional data analysis ties them together. The first chapter describes interactive open-source software for explaining and disseminating results of functional data analyses. Chapters two and three use curve alignment, or registration, to solve common problems in accelerometry and neuroimaging, respectively. The final chapter introduces a novel regression method for modeling functional outcomes that are trajectories over time. The first chapter of this thesis details a software package for interactively visualizing functional data analyses. The software is designed to work for a wide range of datasets and several types of analyses. This chapter describes that software and provides an overview ofFDA in different contexts. The second chapter introduces a framework for curve alignment, or registration, of exponential family functional data. The approach distinguishes itself from previous registration methods in its ability to handle dense binary observations with computational efficiency. Motivation comes from the Baltimore Longitudinal Study on Aging, in which accelerometer data provides valuable insights into the timing of sedentary behavior. The third chapter takes lessons learned about curve registration from the second chapter and use them to develop methods in an entirely new context: large multisite brain imaging studies. Scanner effects in multisite imaging studies are non-biological variability due to technical differences across sites and scanner hardware. This method identifies and removes scanner effects by registering cumulative distribution functions of image intensities values. In the final chapter the focus shifts from curve registration to regression. Described within this chapter is an entirely new nonlinear regression framework that draws from both functional data analysis and systems of ordinary equations. This model is motivated by the neurobiology of skilled movement, and was developed to capture the relationship between neural activity and arm movement in mice.

Regression testing is an expensive software engineering activity intended to provide confidence that modifications to a software system have not introduced faults. Test case prioritization techniques help to reduce regression testing cost by ordering test cases in a way that better achieves testing objectives. In this thesis, we are interested in prioritizing to maximize a test suite's rate of fault detection, measured by a metric, APED, trying to detect regression faults as early as possible during testing. In previous work, several prioritization techniques using low-level code coverage information had been developed. These techniques try to maximize APED over a sequence of software releases, not targeting a particular release. These techniques' effectiveness was empirically evaluated. We present a larger set of prioritization techniques that use information at arbitrary granularity levels and incorporate modification information, targeting prioritization at a particular software release. Our empirical studies show significant improvements in the rate of fault detection over randomly ordered test suites. Previous work on prioritization assumed uniform test costs and fault seventies, which might not be realistic in many practical cases. We present a new cost-cognizant metric, APFD[subscript c], and prioritization techniques, together with approaches for measuring and estimating these costs. Our empirical studies evaluate prioritization in a cost-cognizant environment. Prioritization techniques have been developed independently with little consideration of their similarities. We present a general prioritization framework that allows us to express existing prioritization techniques by a framework algorithm using parameters and specific functions. Previous research assumed that prioritization was always beneficial if it improves the APFD metric. We introduce a prioritization cost-benefit model that more accurately captures relevant cost and benefit factors, and allows practitioners to assess whether it is economical to employ prioritization. Prioritization effectiveness varies across programs, versions, and test suites. We empirically investigate several of these factors on substantial software systems and present a classification-tree-based predictor that can help select the most appropriate prioritization technique in advance. Together, these results improve our understanding of test case prioritization and of the processes by which it is performed. / Graduation date: 2004