published_or_final_version / abstract / Statistics and Actuarial Science / Master / Master of Philosophy

Chan Ho-leung. / "December 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 154-[160]). / 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.

Typescript (photocopy). / Digitized by Kansas Correctional Industries

In order to study the real world, scientists (and computer scientists) develop simplified models that attempt to capture the essential features of the observed system. Understanding the power and limitations of these models, when they apply or fail to fully capture the situation at hand, is therefore of uttermost importance. In this thesis, we investigate the role of some of these models in property testing of probability distributions (distribution testing), as well as in related areas. We introduce natural extensions of the standard model (which only allows access to independent draws from the underlying distribution), in order to circumvent some of its limitations or draw new insights about the problems they aim at capturing. Our results are organized in three main directions: (i) We provide systematic approaches to tackle distribution testing questions. Specifically, we provide two general algorithmic frameworks that apply to a wide range of properties, and yield efficient and near-optimal results for many of them. We complement these by introducing two methodologies to prove information-theoretic lower bounds in distribution testing, which enable us to derive hardness results in a clean and unified way. (ii) We introduce and investigate two new models of access to the unknown distributions, which both generalize the standard sampling model in different ways and allow testing algorithms to achieve significantly better efficiency. Our study of the power and limitations of algorithms in these models shows how these could lead to faster algorithms in practical situations, and yields a better understanding of the underlying bottlenecks in the standard sampling setting. (iii) We then leave the field of distribution testing to explore areas adjacent to property testing. We define a new algorithmic primitive of sampling correction, which in some sense lies in between distribution learning and testing and aims to capture settings where data originates from imperfect or noisy sources. Our work sets out to model these situations in a rigorous and abstracted way, in order to enable the development of systematic methods to address these issues.

The field of property testing has been studied for decades, and Boolean functions are among the most classical subjects to study in this area. In this thesis we consider the property testing of Boolean functions: distinguishing whether an unknown Boolean function has some certain property (or equivalently, belongs to a certain class of functions), or is far from having this property. We study this problem under both the standard setting, where the distance between functions is measured with respect to the uniform distribution, as well as the distribution-free setting, where the distance is measured with respect to a fixed but unknown distribution. We obtain both new upper bounds and lower bounds for the query complexity of testing various properties of Boolean functions: - Under the standard model of property testing, we prove a lower bound of \Omega(n^{1/3}) for the query complexity of any adaptive algorithm that tests whether an n-variable Boolean function is monotone, improving the previous best lower bound of \Omega(n^{1/4}) by Belov and Blais in 2015. We also prove a lower bound of \Omega(n^{2/3}) for adaptive algorithms, and a lower bound of \Omega(n) for non-adaptive algorithms with one-sided errors that test unateness, a natural generalization of monotonicity. The latter lower bound matches the previous upper bound proved by Chakrabarty and Seshadhri in 2016, up to poly-logarithmic factors of n. - We also study the distribution-free testing of k-juntas, where a function is a k-junta if it depends on at most k out of its n input variables. The standard property testing of k-juntas under the uniform distribution has been well understood: it has been shown that, for adaptive testing of k-juntas the optimal query complexity is \Theta(k); and for non-adaptive testing of k-juntas it is \Theta(k^{3/2}). Both bounds are tight up to poly-logarithmic factors of k. However, this problem is far from clear under the more general setting of distribution-free testing. Previous results only imply an O(2^k)-query algorithm for distribution-free testing of k-juntas, and besides lower bounds under the uniform distribution setting that naturally extend to this more general setting, no other results were known from the lower bound side. We significantly improve these results with an O(k^2)-query adaptive distribution-free tester for k-juntas, as well as an exponential lower bound of \Omega(2^{k/3}) for the query complexity of non-adaptive distribution-free testers for this problem. These results illustrate the hardness of distribution-free testing and also the significant role of adaptivity under this setting. - In the end we also study distribution-free testing of other basic Boolean functions. Under the distribution-free setting, a lower bound of \Omega(n^{1/5}) was proved for testing of conjunctions, decision lists, and linear threshold functions by Glasner and Servedio in 2009, and an O(n^{1/3})-query algorithm for testing monotone conjunctions was shown by Dolev and Ron in 2011. Building on techniques developed in these two papers, we improve these lower bounds to \Omega(n^{1/3}), and specifically for the class of conjunctions we present an adaptive algorithm with query complexity O(n^{1/3}). Our lower and upper bounds are tight for testing conjunctions, up to poly-logarithmic factors of n.

Chan Pui Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 103-105). / Abstracts in English and Chinese. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Main Problems --- p.1 / Chapter 1.2 --- Motivation --- p.5 / Chapter 1.3 --- Present Works --- p.7 / Chapter Chapter 2 --- Estimation of the Eigenvalues in a Wishart Distribution --- p.11 / Chapter 2.1 --- Review of Previous Works --- p.14 / Chapter 2.2 --- Some Useful Statistical and Mathematical Results --- p.17 / Chapter 2.3 --- Improved Estimation of A under Squared Error Loss L1 --- p.23 / Chapter 2.4 --- Simulation Study for the Wishart Distribution under Squared Error Loss --- p.28 / Chapter 2.5 --- Discussions on Wishart Distribution Under Squared Error Loss --- p.32 / Chapter 2.6 --- Improved Estimation of A under the Entropy Loss(det) L2 --- p.33 / Chapter 2.7 --- Simulation Study for the Wishart Distribution Under Entropy Loss L2 --- p.38 / Chapter 2.8 --- Discussions on Wishart Distribution Under Entropy Loss --- p.44 / Chapter Chapter 3 --- Estimation of the Eigenvalues in a Multivariate F Distribution --- p.46 / Chapter 3.1 --- Review of Previous Works --- p.49 / Chapter 3.2 --- Some Useful Statistical and Mathematical Results --- p.50 / Chapter 3.3 --- Improved Estimation of A under the Squared Loss L1 --- p.54 / Chapter 3.4 --- Simulation Study for F Distribution under Squared Error Loss L1 --- p.62 / Chapter 3.5 --- Discussions on F distribution under Squared Error Loss --- p.68 / Chapter 3.6 --- Improved Estimation of A under the Entropy Loss(det) L2 --- p.69 / Chapter 3.7 --- Simulation Study for Multivariate F Distribution under Entropy Loss(det) L2 --- p.76 / Chapter 3.8 --- Discussions on F distribution under Entropy Loss --- p.86 / Chapter Chapter 4 --- Inheritance of Dominance between Eigenvalues Loss Function and Matrix Function --- p.87 / Chapter 4.1 --- Significance of The Problem --- p.87 / Chapter 4.2 --- Inheritance of Dominance between Eigenvalues Estimator and Matrix Estimator under Squared Error Loss --- p.92 / Chapter 4.3 --- Inheritance of Dominance between Eigenvalues Estimator and Matrix Estimator under Entropy Loss --- p.97 / Chapter 4.4 --- Conclusion --- p.102 / BIBLIOGRAPHY --- p.103

When sampling from multivariate distributions whose density contours are banana-shaped due to the non-linear correlation structure, traditional Markov chain Monte Carlo methods such as random walk Metropolis and independent Metropolis-Hastings suffer from severe low convergence. In this thesis, a model for bivariate banana-shaped distributions is constructed which is used to fit general banana-shaped distributions in terms of the probability density function. Transformations which are aimed at converting the variables to orthogonal variables by changing the coordinate system are designed for this distribution model. A new Markov chain Monte Carlo algorithm involving this set of transformations is proposed to sample these complex distributions. The key point of the new algorithm is to approximate the target density function using function using a parametric model which can facilitate the MCMC sampling after changing to another coordinate system. Detailed comparisons of convergence rate and estimation efficiency between the new method and existing methods are performed using both benchmark examples and practical examples, which showed the advantage of the new method. / 在多元概率分佈中，如果變量問存在非線性相關性使其等高線為香蕉形，傳統的馬爾科夫蒙特卡洛方法，如隨機漫步蒙特卡洛及獨立M-H方法都只有非常低的收斂速度和有效樣本數。本論文設計一種二元香蕉形分佈函數模型對一般香蕉形分佈進行擬合及一套能將其變量正交化的變換函數，並以此模型及變換函數為基礎建立一種新的馬爾科夫蒙特卡洛方法，實現對香蕉形分佈的高效率抽樣。該方法的關鍵在於對一般香蕉形分佈進行近似的參數模型能夠在進行坐標轉換後便於採樣。本論文將在不同例子中以收斂速度及估計效率為標準比較新方法與已有方法，模擬實驗和實例都顯示新方法較優。 / Chan, Kwun Cheung. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 50-52). / Abstracts also in Chinese. / Title from PDF title page (viewed on 05, October, 2016). / Detailed summary in vernacular field only.

There has been widespread concern expressed by members of the statistics education community in the past few years about the lack of any real understanding demonstrated by many students completing courses in introductory statistics. This deficiency in understanding has been particularly noted in the area of inferential statistics, where students, particularly those studying statistics as a service course, have been inclined to view statistical inference as a set of unrelated recipes. As such, these students have developed skills that have little practical application and are easily forgotten. This thesis is concerned with the development of understanding in statistical inference for beginning students of statistics at the post-secondary level. This involves consideration of the nature of understanding in introductory statistical inference, and how understanding can be measured in the context of statistical inference. In particular, the study has examined the role of the sampling distribution in the students? schemas for statistical inference, and its relationship to both conceptual and procedural understanding. The results of the study have shown that, as anticipated, students will construct highly individual schemas for statistical inference but that the degree of integration of the concept of sampling distribution within this schema is indicative of the level of development of conceptual understanding in that student. The results of the study have practical implications for the teaching of courses in introductory statistics, in terms of content, delivery and assessment.

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 83-85).

Thesis (Ph. D.)--Florida State University, 2004. / Advisor: Dr. Craig A. Nolder, Florida State University, College of Arts and Sciences, Dept. of Mathematics. Title and description from dissertation home page (viewed June 29, 2004). Includes bibliographical references.