Digitized by Kansas Correctional Industries

Ng, Ka Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 45-47). / Abstracts in English and Chinese. / Introduction --- p.v / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Noncommutative probability spaces and Free in- dependence --- p.1 / Chapter 1.2 --- C*-probability spaces --- p.4 / Chapter 1.3 --- Fock spaces --- p.5 / Chapter 1.4 --- Cauchy transform and R-transform of probability measures with bounded support --- p.7 / Chapter 1.5 --- Helton-Howe formula --- p.8 / Chapter 1.6 --- Stieltjes inversion formula --- p.9 / Chapter 1.7 --- Pick functions --- p.11 / Chapter 2 --- Free convolution and R-transform --- p.13 / Chapter 2.1 --- Additive free convolution and R-transform --- p.13 / Chapter 2.2 --- R-transform and algebraic Cauchy transform --- p.18 / Chapter 2.3 --- Properties of R-transform --- p.26 / Chapter 2.4 --- Properties of --- p.29 / Chapter 3 --- Examples of free convolution --- p.32 / Chapter 3.1 --- Measures with compact support --- p.32 / Chapter 3.2 --- Examples of free convolution --- p.33 / Chapter 4 --- Free Central Limit Theorem --- p.42 / Bibliography --- p.45

In mathematics and statistics, the desire to eliminate mathematical tedium and facilitate exploration has lead to computer algebra systems. These computer algebra systems allow students and researchers to perform more of their work at a conceptual level. The design of generic algorithms for tedious computations allows modelers to push current modeling boundaries outward more quickly.;Probability theory, with its many theorems and symbolic manipulations of random variables is a discipline in which automation of certain processes is highly practical, functional, and efficient. There are many existing statistical software packages, such as SPSS, SAS, and S-Plus, that have numeric tools for statistical applications. There is a potential for a probability package analogous to these statistical packages for manipulation of random variables. The software package being developed as part of this dissertation, referred to as "A Probability Programming Language" (APPL) is a random variable manipulator and is proposed to fill a technology gap that exists in probability theory.;My research involves developing algorithms for the manipulation of discrete random variables. By defining data structures for random variables and writing algorithms for implementing common operations, more interesting and mathematically intractable probability problems can be solved, including those not attempted in undergraduate statistics courses because they were deemed too mechanically arduous. Algorithms for calculating the probability density function of order statistics, transformations, convolutions, products, and minimums/maximums of independent discrete random variables are included in this dissertation.

Let X₁, X2, ••• , Xn be n independent and identically distributed random variables having the unity-truncated logarithmic series distribution with probability function given by f(x;0) = ᵅθX ⁄ x x ε T where α = [ -log(1-θ) -θ ] 0 < θ < 1, and T = {2,3,…,∞}. Define their sum as Z = X₁ + X2 + … + Xn . We derive here the distribution of Z, denoted by p(z;n,θ), using the inversion formula for characteristic functions, in an explicit form in terms of a linear combination of Stirling numbers of the first kind. A recurrence relation for the probability function p(z;n,θ) is obtained and is utilized to provide a short table of pCz;n,8) for certain values of n and θ. Furthermore, some properties of p(z;n,θ) are investigated following Patil and Wani [Sankhla, Series A, 27, (1965), 27l-280J.

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High-dimensional data offers researchers increased ability to find useful factors in predicting a response. However, determination of the most important factors requires careful selection of the explanatory variables. In order to tackle this challenge, much work has been done on single or grouped variable selection under the penalized regression framework. Although the topic of variable selection has been extensively studied under the parametric framework, its applications to more flexible nonparametric models are yet to be explored. In order to implement the variable selection in nonparametric additive models, I introduce and study two nonconvex selection methods under the penalized regression framework, namely the group MCP and the adaptive group LASSO, aiming at improvements on the selection performances of the more widely known group LASSO method in such models. One major part of the dissertation focuses on the theoretical properties of the group MCP and the adaptive group LASSO. I derive their selection and estimation properties. The application of the presently proposed methods to nonparametric additive models are further examined using simulation. Their applications to areas such as the economics and genomics are presented as well. Under both the simulation studies and data applications, the group MCP and the adaptive group LASSO have shown their advantages over the more traditionally used group LASSO method. For the proposed adaptive group LASSO that uses the newly proposed weights, whose recursive application is therefore never studied before, I also derive its theoretical properties under a very general framework. Simulation studies under linear regression are included. In addition to the theoretical and empirical investigations, throughout the dissertation, several other important issues have been briefly discussed, including the computing algorithms and different ways of selecting tuning parameters.

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1v. (various paging) : / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (D.Sc.)--University of Adelaide, Dept. of Mathematical Sciences, 1981

Thesis (Ph. D.)--Oregon State University, 1969. / Typescript. Includes bibliographical references (leaf 43). Also available on the World Wide Web.

Includes bibliographical references