M.Sc. (Mathematical Statistics) / While a typical regression model describes how the mean value of a response variable varies with a set of explanatory variables, quantile regression describes the variation in the quantiles of the response. When the response distribution di ers substantially from normality the quantiles provide a substantially richer description of the distribution than can be obtained by standard regression, and is obtainable without making any assumptions on the form of the underlying distribution. In this dissertation we study the theory of quantile regression models, with particular focus on the application of quantile regression methods to censored survival data. While the statistical literature on censored quantile regression methods is extensive, the computational di culties and complicated inferential and asymptotic arguments associated with many of these approaches present a considerable stumbling block in the routine application of the methodology. We discuss in detail a more recent approach which is based on counting processes and martingale properties associated with counting processes. The inferential and asymptotic properties of this method provides some notable advantages over comparable methods. The performance of the method is examined using Monte Carlo Simulation, as well as an application to a large loan portfolio of a nancial institution.

University of Macau / Faculty of Science and Technology. / Department of Mathematics

by Wai-leung Li. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 74-77). / Chapter Chapter 1 --- Introduction --- p.1 / Chapter § 1.1 --- Why transformation of variables in regression analysis is needed? --- p.1 / Chapter § 1.2 --- Suggested functional transformation -- Box-Cox Transformation --- p.3 / Chapter § 1.3 --- Methodology --- p.5 / Chapter § 1.4 --- General theory of constructing asymptotic confidence intervals and confidence regions --- p.9 / Chapter § 1.4.1 --- Method based on the log-likelihood ratio statistic --- p.9 / Chapter § 1.4.2 --- Method based on the asymptotic normality of the maximum likelihood estimates --- p.13 / Chapter § 1.4.3 --- Method based on the score statistic --- p.15 / Chapter § 1.5 --- General theory of constructing exact confidence intervals and confidence regions --- p.17 / Chapter § 1.6 --- Summary --- p.23 / Chapter Chapter 2 --- Confidence Intervals for the non-linear parameter λ in the Box-Cox transformation models --- p.24 / Chapter § 2.1 --- Confidence intervals based on the log-likelihood ratio statistics --- p.26 / Chapter § 2.1.1 --- Asymptotically equivalent forms for constructing confidence intervals based on the log-likelihood ratio statistics --- p.30 / Chapter § 2.2 --- Confidence intervals based on the asymptotic normality of the maximum likelihood estimates --- p.31 / Chapter § 2.3 --- Confidence intervals based on the score statistics --- p.35 / Chapter § 2.4 --- Confidence intervals based on the exact test --- p.36 / Chapter § 2.5 --- Small simulation studies of constructing confidence intervals for A based on the four different methods --- p.37 / Chapter § 2.5.1 --- Design of the simulation studies --- p.40 / Chapter § 2.5.2 --- Simulation results --- p.41 / Chapter § 2.6 --- Summary --- p.44 / Chapter Chapter 3 --- Confidence Regions for the parameters in the Box-Cox transformation models --- p.45 / Chapter § 3.1 --- Confidence regions based on the log-likelihood ratio statistics --- p.45 / Chapter § 3.1.1 --- "Confidence region for (λ,ζ1）based on the log-likelihood ratio statistics" --- p.46 / Chapter § 3.1.2 --- Confidence region for (ζ1）based on the log-likelihood ratio statistics --- p.51 / Chapter § 3.2 --- Confidence regions based on the asymptotic normality of the maximum likelihood estimates --- p.53 / Chapter § 3.2.1 --- "Confidence region for (λ,ζ1）based on the asymptotic normality of the maximum likelihood estimates" --- p.53 / Chapter § 3.2.2 --- Confidence region for (ζ1）based on the asymptotic normality of the maximum likelihood estimates --- p.57 / Chapter § 3.3 --- Confidence regions based on the score statistics --- p.58 / Chapter § 3.3.1 --- "Confidence region for (λ,ζ1） based on the score statistic" --- p.59 / Chapter § 3.3.2 --- Confidence region for (ζ1 ) based on the score statistic --- p.60 / Chapter § 3.4 --- Confidence region based on the exact test --- p.61 / Chapter § 3.5 --- Small simulation studies of constructing confidence regions for the parameters of interest based on the four different methods --- p.62 / Chapter Chapter 4 --- Robustness and Discussion --- p.67 / Chapter §4.1 --- Contamination normal distribution --- p.67 / Chapter § 4.1.1 --- Confidence intervals for the non- linear parameter λ based on the contamination normal distribution of error terms --- p.68 / Chapter § 4.1.2 --- Confidence regions for the parameters of interest based on the contamination normal distribution of the error terms --- p.70 / Chapter § 4.2 --- Summary --- p.72 / References --- p.74 / Figures / Appendix A / Appendix B / Appendix C / Appendix D

by Maggie Wai Ling Poon. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 70-78). / Abstract and questionnaire also in Chinese. / Chapter 1. --- ABSTRACT --- p.3 / Chapter 2. --- ACKNOWLEDGEMENTS --- p.4 / Chapter 3. --- INTRODUCTION --- p.5 / Cognitive factors / The content specificity hypothesis / Empirical support of the cognitive model / Purpose of the present study / Chapter 4. --- METHOD --- p.21 / Subjects / Measures / Chapter 5. --- RESULTS --- p.28 / Psychometric properties / Correlational analysis / Hierarchical regression analyses / Chapter 6. --- DISCUSSION --- p.53 / Psychometric properties of instruments / Empirical support to the cognitive model / Implications of the study / Summary and conclusion / Comments and future direction / Chapter 7. --- REFERENCES --- p.70 / Chapter 8. --- APPENDICES --- p.79

Digitized by Kansas Correctional Industries

Typescript (photocopy). / Digitized by Kansas Correctional Industries

Wenyang Zhang. / "October 1998." / Thesis (Ph.D.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (p. 190-196). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

In many application areas ranging from bioinformatics to imaging, we are interested in recovering a sparse coefficient in the high-dimensional linear model, when the sample size n is comparable to or less than the dimension p. One of the most popular classes of estimators is the Lq-regularized least squares (LQLS), a.k.a. bridge regression. There have been extensive studies towards understanding the performance of the best subset selection (q=0), LASSO (q=1) and ridge (q=2), three widely known estimators from the LQLS family. This thesis aims at giving a unified view of LQLS for all the non-negative values of q. In contrast to most existing works which obtain order-wise error bounds with loose constants, we derive asymptotically exact error formulas characterized through a series of fixed point equations. A delicate analysis of the fixed point equations enables us to gain fruitful insights into the statistical properties of LQLS across the entire spectrum of Lq-regularization. Our work not only validates the scope of folklore understanding of Lq-minimization, but also provides new insights into high-dimensional statistics as a whole. We will elaborate on our theoretical findings mainly from parameter estimation point of view. At the end of the thesis, we briefly mention bridge regression for variable selection and prediction. We start by considering the parameter estimation problem and evaluate the performance of LQLS by characterizing the asymptotic mean square error (AMSE). The expression we derive for AMSE does not have explicit forms and hence is not useful in comparing LQLS for different values of q, or providing information in evaluating the effect of relative sample size n/p or the sparsity level of the coefficient. To simplify the expression, we first perform the phase transition (PT) analysis, a widely accepted analysis diagram, of LQLS. Our results reveal some of the limitations and misleading features of the PT framework. To overcome these limitations, we propose the small-error analysis of LQLS. Our new analysis framework not only sheds light on the results of the phase transition analysis, but also describes when phase transition analysis is reliable, and presents a more accurate comparison among different Lq-regularizations. We then extend our low noise sensitivity analysis to linear models without sparsity structure. Our analysis, as a generalization of phase transition analysis, reveals a clear picture of bridge regression for estimating generic coefficients. Moreover, by a simple transformation we connect our low-noise sensitivity framework to the classical asymptotic regime in which n/p goes to infinity, and give some insightful implications beyond what classical asymptotic analysis of bridge regression can offer. Furthermore, following the same idea of the new analysis framework, we are able to obtain an explicit characterization of AMSE in the form of second-order expansions under the large noise regime. The expansions provide us some intriguing messages. For example, ridge will outperform LASSO in terms of estimating sparse coefficients when the measurement noise is large. Finally, we present a short analysis of LQLS, for the purpose of variable selection and prediction. We propose a two-stage variable selection technique based on the LQLS estimators, and describe its superiority and close connection to parameter estimation. For prediction, we illustrate the intricate relation between the tuning parameter selection for optimal in-sample prediction and optimal parameter estimation.

No description available.

Lai, Chu Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 44-46). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- B-Spline Modeling of Nonparametric Fixed Effects --- p.3 / Chapter 3 --- Parameter Estimation --- p.5 / Chapter 3.1 --- Fixed Component Estimation using Adaptive Group Lasso --- p.5 / Chapter 3.2 --- Random Component Estimation using Newton Raphson --- p.7 / Chapter 3.3 --- Combining the Two Algorithms --- p.9 / Chapter 4 --- Selection of Model Complexity --- p.10 / Chapter 4.1 --- Model Selection Criterion --- p.10 / Chapter 4.2 --- Calculating the Degrees of Freedom --- p.10 / Chapter 4.3 --- Practical Minimization of (4.1) --- p.12 / Chapter 5 --- Theoretical results / Chapter 5.1 --- Consistency of adaptive group lasso --- p.14 / Chapter 5.2 --- Consistency of Bayesian Information Criterion --- p.16 / Chapter 6 --- Simulations / Chapter 7 --- Real applications / Chapter 7.1 --- Prostate cancer data --- p.23 / Chapter 7.2 --- Housing data --- p.25 / Chapter 7.3 --- Depression Dataset --- p.27 / Chapter 8 --- Summary --- p.31 / Chapter A --- Derivation of (3.7) and (3.8) --- p.32 / Chapter B --- Lemmas --- p.34 / Chapter C --- Proofs of theorems --- p.37