Thesis (Ph. D.) -- California Institute of Technology, 2010. / Title from home page (viewed 06/21/2010). Advisor and committee chair names found in the thesis' metadata record in the digital repository. Includes bibliographical references.

For an inequality system, an error bound is an estimation for the distance from any point to the solution set of the inequality. The Ekeland variational principle (EVP) is an important tool in the study of error bounds. We prove that EVP is equivalent to an error bound result and present several sufficient conditions for an inequality system to have error bounds. In a metric space, a condition is similar to that of Takahashi. In a Banach space we express conditions in terms of an abstract subdifferential and the lower Dini derivative. We then discuss error bounds with exponents by a relation between the lower Dini derivatives of a function and its power function. For an l.s.c. convex function on a reflexive Banach space these conditions turn out to be equivalent. Furthermore a global error bound closely relates to the metric regularity. / Graduate

D.Phil. / none / Prof. W.F. Harris

by Lai Siu Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1989. / Bibliography: leaves 50-52.

Zhang Yuanxiu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 38-39). / Abstracts in English and Chinese.

The overall theme of this thesis work concerns the problem of handling measurement error and sources of variation in functional data models. The first part introduces a wavelet-based sparse principal component analysis approach for characterizing the variability of multilevel functional data that are characterized by spatial heterogeneity and local features. The total covariance of the data can be decomposed into three hierarchical levels: between subjects, between sessions and measurement error. Sparse principal component analysis in the wavelet domain allows for reducing dimension and deriving main directions of random effects that may vary for each hierarchical level. The method is illustrated by application to data from a study of human vision. The second part considers the problem of scalar-on-function regression when the functional regressors are observed with measurement error. We develop a simulation-extrapolation method for scalar-on-function regression, which first estimates the error variance, establishes the relationship between a sequence of added error variance and the corresponding estimates of coefficient functions, and then extrapolates to the zero-error. We introduce three methods to extrapolate the sequence of estimated coefficient functions. In a simulation study, we compare the performance of the simulation-extrapolation method with two pre-smoothing methods based on smoothing splines and functional principal component analysis. The third part discusses several extensions of the simulation-extrapolation method developed in the second part. Some of the extensions are illustrated by application to diffusion tensor imaging data.

Li Guo-Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 105-107). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Examples for the variational inequality problem --- p.2 / Chapter 1.2 --- Approaches for variational inequality problem --- p.7 / Chapter 1.3 --- Error bounds results for variational inequality problem --- p.8 / Chapter 1.4 --- Organization --- p.9 / Chapter 2 --- Solution Theory --- p.11 / Chapter 2.1 --- "Elementary Convex Analysis, Nonsmooth Analysis and Degree theory" --- p.11 / Chapter 2.1.1 --- Elementary Convex Analysis --- p.11 / Chapter 2.1.2 --- Elementary Nonsmooth Analysis --- p.16 / Chapter 2.1.3 --- Degree Theory --- p.18 / Chapter 2.2 --- Existence and Uniqueness Theory --- p.24 / Chapter 3 --- Merit Functions for variational inequalities problem --- p.36 / Chapter 3.1 --- Regularized gap function --- p.38 / Chapter 3.2 --- D-gap function --- p.44 / Chapter 3.3 --- Generalized Regularize gap function and Generalized D-gap function --- p.61 / Chapter 4 --- Error bound results for the merit functions --- p.74 / Chapter 4.1 --- Error bound results for Regularized gap function --- p.77 / Chapter 4.2 --- Error bound results for D-gap function --- p.78 / Chapter 4.3 --- Error bound results for Generalized Regularized gap function --- p.92 / Chapter 4.4 --- Error bound results for Generalized D-gap function --- p.93 / Bibliography --- p.105

Wang Xin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 61-63). / Abstracts in English and Chinese. / Chapter Chapter 1 --- Introduction and Literature Review --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- The Objective --- p.2 / Chapter 1.3 --- Literature Review --- p.3 / Chapter Chapter 2 --- The Model --- p.7 / Chapter Chapter 3 --- The Consistent Estimation of Chang-Point --- p.11 / Chapter 3.1 --- The Consistency of ιT --- p.11 / Chapter 3.2 --- The Case of Single Regressor --- p.15 / Chapter Chapter 4 --- The SupWald Test and Sample-Splitting Method --- p.20 / Chapter 4.1 --- The SupWald Test --- p.20 / Chapter 4.2 --- " The Distribution of WT (ι,0,1)" --- p.23 / Chapter 4.3 --- The Sample-Splitting Method --- p.25 / Chapter Chapter 5 --- Experimental Results --- p.27 / Chapter Chapter 6 --- Empirical Example --- p.41 / Chapter Chapter 7 --- Conclusion --- p.44 / Appendix --- p.46 / References --- p.61

Chan Ka Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 46-48). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Multiple comparison procedures and their applications --- p.1 / Chapter 1.2 --- Different types of error control --- p.3 / Chapter 1.3 --- Single-step and stepwise procedures --- p.5 / Chapter 1.4 --- From familywise error rate control to false discovery rate control --- p.8 / Chapter 1.5 --- The FDR procedure of BH --- p.10 / Chapter 1.6 --- Application of the FDR procedure --- p.11 / Chapter 1.7 --- Family size and family size robustness --- p.16 / Chapter 1.8 --- Objectives of the thesis --- p.17 / Chapter 2 --- The Familywise Robustness Criteria --- p.18 / Chapter 2.1 --- The basic idea of familywise robustness --- p.18 / Chapter 2.2 --- Definitions and notations --- p.19 / Chapter 2.3 --- The measurement of robustness to changing family size --- p.21 / Chapter 2.4 --- Main Theorems --- p.21 / Chapter 2.5 --- Example --- p.23 / Chapter 2.6 --- Summary --- p.24 / Chapter 3 --- FDR and FWR --- p.26 / Chapter 3.1 --- Positive false discovery rate --- p.26 / Chapter 3.2 --- A unified approach to FDR --- p.29 / Chapter 3.3 --- The S procedure --- p.30 / Chapter 3.4 --- Family wise robustness criteria and the S procedure --- p.32 / Chapter 4 --- Simulation Study --- p.41 / Chapter 4.1 --- The setup --- p.41 / Chapter 4.2 --- Simulation result --- p.43 / Chapter 4.3 --- Conclusions --- p.44 / Bibliography --- p.46

Zheng Xi Yin. / "December 2002." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (p. 117-121). / 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 Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.