published_or_final_version / Statistics / Doctoral / Doctor of Philosophy

Yung Hon-wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 133-139). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.1 / Chapter 1 --- Density Theorems in Vector Optimization --- p.7 / Chapter 1.1 --- Preliminary --- p.7 / Chapter 1.2 --- The Arrow-Barankin-Blackwell Theorem in Normed Spaces --- p.14 / Chapter 1.3 --- The Arrow-Barankin-Blackwell Theorem in Topolog- ical Vector Spaces --- p.27 / Chapter 1.4 --- Density Results in Dual Space Setting --- p.32 / Chapter 2 --- Density Theorem for Super Efficiency --- p.45 / Chapter 2.1 --- Definition and Criteria for Super Efficiency --- p.45 / Chapter 2.2 --- Henig Proper Efficiency --- p.53 / Chapter 2.3 --- Density Theorem for Super Efficiency --- p.58 / Chapter 3 --- Connectedness Results in Vector Optimization --- p.63 / Chapter 3.1 --- Set-valued Maps --- p.64 / Chapter 3.2 --- The Contractibility of the Efficient Point Sets --- p.67 / Chapter 3.3 --- Connectedness Results in Vector Optimization Prob- lems --- p.83 / Chapter 4 --- Error Bounds In Normed Spaces --- p.90 / Chapter 4.1 --- Error Bounds of Lower Semicontinuous Functionsin Normed Spaces --- p.91 / Chapter 4.2 --- Error Bounds of Lower Semicontinuous Convex Func- tions in Reflexive Banach Spaces --- p.100 / Chapter 4.3 --- Error Bounds with Fractional Exponents --- p.105 / Chapter 4.4 --- An Application to Quadratic Functions --- p.114 / Bibliography --- p.133

First, we study the associated regularized gap functions and the D-gap functions and compute their Clarke-Rockafellar directional derivatives and the Clarke generalized gradients. Second, using these tools and extending the works of Fukushima and Pang (who studied the case when F is smooth), we present results on the relationship between minimizing sequences and stationary sequences of the D-gap functions, regardless the existence of solutions of (VIP). Finally, as another application, we show that, under the strongly monotonicity assumption, the regularized gap functions have fractional exponent error bounds, and thereby we provide an algorithm of Armijo type to solve the (VIP). / In this thesis, we investigate a nonsmooth variational inequality problem (VIP) defined by a locally Lipschitz function F which is not necessarily differentiable or monotone on its domain which is a closed convex set in an Euclidean space. / Tan Lulin. / "December 2005." / Adviser: Kung Fu Ng. / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6444. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 79-84) and index. / 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.

Four-dimensional variational data assimilation (4D-Var) provides an estimate to the state of a dynamical system through the minimization of a cost functional that measures the distance to a prior state (background) estimate and observations over a time window. The analysis fit to each information input component is determined by the specification of the error covariance matrices in the data assimilation system (DAS). Weak-constraint 4D-Var (w4D-Var) provides a theoretical framework to account for modeling errors in the analysis scheme. In addition to the specification of the background error covariance matrix, the w4D-Var formulation requires information on the model error statistics and specification of the model error covariance. Up to now, the increased computational cost associated with w4D-Var has prevented its practical implementation. Various simplifications to reduce the computational burden have been considered, including writing the model error covariance as a scalar multiple of the background error covariance and modeling the model error. In this thesis, the main objective is the development of computationally feasible techniques for the improved representation of the model error statistics in a data assimilation system. Three new approaches are considered. A Monte Carlo method that uses an ensemble of w4D-Var systems to obtain flow-dependent estimates to the model error statistics. The evaluation of statistical diagnostic equations involving observation residuals to estimate the model error covariance matrix. An adaptive tuning procedure based on the sensitivity of a short-range forecast error measure to the model error DAS parametrization. The validity and benefits of these approaches are shown in two stages of numerical experiments. A proof-of-concept is shown using the Lorenz multi-scale model and the shallow water equations for a one-dimensional domain. The results show the potential of these methodologies to produce improved state estimates, as compared to other approaches in data assimilation. It is expected that the techniques presented will find an extended range of applications to assess and improve the performance of a w4D-Var system.

The accurate determination of cloud cover amount is important for characterizing the role of cloud feedbacks in the climate system. Clouds have a large influence on the climate system through their effect on the earth's radiation budget. As indicated by the NASA Earth Radiation Budget Experiment (ERBE), the change in the earth's radiation budget brought about by clouds is ~-15 Wm⁻² on a global scale, which is several times the ~4 Wm⁻² gain in energy to the troposphere-surface system that would arise from a doubling of CO₂ in the atmosphere. Consequently, even a small change in global cloud amount may lead to a major change in the climate system. Threshold methods are commonly used to derive cloud properties from satellite imagery data. Here, in order to quantify errors due to thresholds, cloud cover is obtained using three different values of thresholds. The three thresholds are applied to the 11 μm, (4 km)² NOAA-9 AVHRR GAC satellite imagery data over four oceanic regions. Regional cloud-cover fractions are obtained for two different scales, (60 km)² and (250 km)². The spatial coherence method for obtaining cloud cover from imagery data is applied to coincident data. The differences between cloud cover derived by the spatial coherence method and by the threshold methods depends on the setting of the threshold. Because the spatial coherence method is believed to provide good estimates of cloud cover for opaque, single-layered cloud systems, this study is limited to such systems, and the differences in derived cloud cover are interpreted as errors due to the application of thresholds. The threshold errors are caused by pixels that are partially covered by clouds and the errors have a dependence on the regional scale cloud cover. The errors can be derived from the distribution of pixel-scale cloud cover. Two simple models which assume idealized distributions for pixel-scale cloud cover are constructed and used to estimate the threshold errors. The results show that these models, though simple, perform rather well in estimating the differences between cloud cover derived by the spatial coherence method and those obtained by threshold methods. / Graduation date: 1992

Semiparametric maximum likelihood analysis allows inference in errors-invariables models with small loss of efficiency relative to full likelihood analysis but with significantly weakened assumptions. In addition, since no distributional assumptions are made for the nuisance parameters, the analysis more nearly parallels that for usual regression. These highly desirable features and the high degree of modelling flexibility permitted warrant the development of the approach for routine use. This thesis does so for the special cases of linear and nonlinear regression with measurement errors in one explanatory variable. A transparent and flexible computational approach is developed, the analysis is exhibited on some examples, and finite sample properties of estimates, approximate standard errors, and likelihood ratio inference are clarified with simulation. / Graduation date: 2001

The standard way of representing numbers on computers gives rise to errors which increase as computations progress. Using p-adic valuations can reduce error accumulation. Valuation theory tells us that p-adic and standard valuations cannot be directly compared. The p-adic valuation can, however, be used in an indirect way. This gives a method of doing arithmetic on a subset of the rational numbers without any error. This exactness is highly desirable, and can be used to solve certain kinds of problems which the standard valuation cannot conveniently handle. Programming a computer to use these p-adic numbers is not difficult, and in fact uses computer resources similar to the standard floating-point representation for real numbers. This thesis develops the theory of p-adic valuations, discusses their implementation, and gives some examples where p-adic numbers achieve better results than normal computer computation. / Graduation date: 1994

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 135-139).

In regression analysis, random errors in an explanatory variable cause the usual estimates of its regression coefficient to be biased. Although this problem has been studied for many years, routine methods have not emerged. This thesis investigates some aspects of this problem in the setting of analysis of epidemiological data. A major premise is that methods to cope with this problem must account for the shape of the frequency distribution of the true covariable, e.g., exposure. This is not widely recognized, and many existing methods focus only on the variability of the true covariable, rather than on the shape of its distribution. Confusion about this issue is exacerbated by the existence of two classical models, one in which the covariable is a sample from a distribution and the other in which it is a collection of fixed values. A unified approach is taken here, in which for the latter of these models more attention than usual is given to the frequency distribution of the fixed values. In epidemiology the distribution of exposures is often very skewed, making these issues particularly important. In addition, the data sets can be very large, and another premise is that differences in the performance of methods are much greater when the samples are very large. Traditionally, methods have largely been evaluated by their ability to remove bias from the regression estimates. A third premise is that in large samples there may be various methods that will adequately remove the bias, but they may differ widely in how nearly they approximate the estimates that would be obtained using the unobserved true values. A collection of old and new methods is considered, representing a variety of basic rationales and approaches. Some comparisons among them are made on theoretical grounds provided by the unified model. Simulation results are given which tend to confirm the major premises of this thesis. In particular, it is shown that the performance of one of the most standard approaches, the "correction for attenuation" method, is poor relative to other methods when the sample size is large and the distribution of covariables is skewed. / Graduation date: 1993

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 89-94).