在日常生活和科学研究中产生大量偏好数据，其反应一组被关注对象受偏好的程度。通常用排序数据或多元选择数据来记录观察结果。有时候关于两个对象的偏好没有明显强弱之分，导致排序产生节点，也就是所谓的删失排序。为了研究带有删失的排序数据，基于Thurstone的随机效用假设理论我们建立了一个对称贝叶斯probit模型。然而，参数识别是probit模型必须解决的问题，即确定一组潜在效用的位置和尺度。通常方法是选择其中一个对象为基，然后用其它对象的效用减去这个基的效用，最后我们关于这些效用差来建模。问题是，在用贝叶斯方法处理多元选择数据时，其预测结果对基的选择有敏感性，即选不同对象为基预测结果是不一样的。本文，我们虚构一个基，即一组对象偏好的平均。依靠这个基，我们为多元选择probit模型给出一个不依赖于对象标号的识别方法，即对称识别法。进一步，我们设计一种贝叶斯算法来估计这个模型。通过仿真研究和真实数据分析，我们发现这个贝叶斯probit模型被完全识别，而且消除通常识别法所存在的敏感性。接下来，我们把这个关于多元选择数据建立的probit模型推广到处理一般删失排序数据，即得到对称贝叶斯删失排序probit 模型。最后，我们用这个模型很好的分析了香港赌马数据。 / Vast amount of preference data arise from daily life or scientific research, where observations consist of preferences on a set of available objects. The observations are usually recorded by ranking data or multinomial data. Sometimes, there is not a clear preference between two objects, which will result in ranking data with ties, also called censored rank-ordered data. To study such kind of data, we develop a symmetric Bayesian probit model based on Thurstone's random utility (discriminal process) assumption. However, parameter identification is always an unavoidable problem for probit model, i.e., determining the location and scale of latent utilities. The standard identification method need to specify one of the utilities as a base, and then model the differences of the other utilities subtracted by the base. However, Bayesian predictions have been verified to be sensitive to specification of the base in the case of multinomial data. In this thesis, we set the average of the whole set of utilities as a base which is symmetric to any relabeling of objects. Based on this new base, we propose a symmetric identification approach to fully identify multinomial probit model. Furthermore, we design a Bayesian algorithm to fit that model. By simulation study and real data analysis, we find that this new probit model not only can be identifed well, but also remove sensitivities mentioned above. In what follows, we generalize this probit model to fit general censored rank-ordered data. Correspondingly, we get the symmetric Bayesian censored rank-ordered probit model. At last, we apply this model to analyze Hong Kong horse racing data successfully. / Detailed summary in vernacular field only. / Pan, Maolin. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 50-55). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Overview --- p.2 / Chapter 1.1.1 --- The Ranking Model --- p.2 / Chapter 1.1.2 --- Discrete Choice Model --- p.4 / Chapter 1.2 --- Methodology --- p.7 / Chapter 1.2.1 --- Data Augmentation --- p.8 / Chapter 1.2.2 --- Marginal Data Augmentation --- p.8 / Chapter 1.3 --- An Outline --- p.9 / Chapter 2 --- Bayesian Multinomial Probit Model Based On Symmetric I-denti cation --- p.11 / Chapter 2.1 --- Introduction --- p.11 / Chapter 2.2 --- The MNP Model --- p.14 / Chapter 2.3 --- Symmetric Identification and Bayesian Analysis --- p.17 / Chapter 2.3.1 --- Symmetric Identification --- p.18 / Chapter 2.3.2 --- Bayesian Analysis --- p.21 / Chapter 2.4 --- Case Studies --- p.25 / Chapter 2.4.1 --- Simulation Study --- p.25 / Chapter 2.4.2 --- Clothes Detergent Purchases Data --- p.27 / Chapter 2.5 --- Summary --- p.29 / Chapter 3 --- Symmetric Bayesian Censored Rank-Ordered Probit Model --- p.30 / Chapter 3.1 --- Introduction --- p.30 / Chapter 3.2 --- Ranking Model --- p.33 / Chapter 3.2.1 --- Ranking Data --- p.33 / Chapter 3.2.2 --- Censored Rank-Ordered Probit Model --- p.35 / Chapter 3.2.3 --- Symmetrically Identified CROP Model --- p.36 / Chapter 3.3 --- Bayesian Analysis on Symmetrically Identified CROP Model --- p.37 / Chapter 3.3.1 --- Model Estimation --- p.38 / Chapter 3.4 --- Application: Hong Kong Horse Racing --- p.41 / Chapter 3.5 --- Summary --- p.44 / Chapter 4 --- Conclusion and Further Studies --- p.45 / Chapter A --- Prior for covariance matrix with trace augmented restriction --- p.47 / Chapter B --- Derivation of sampling intervals --- p.49 / Bibliography --- p.50

Rank minimization with affine constraints has various applications in different area. Due to the intractability of rank function in the objective, many alternative functions have been widely studied in the literature, e.g., nuclear norm, and have been shown an effective way both theoretically and practically. The intractability was bypassed as those functions hold some nice properties such as convexity and differentiability. In this dissertation, we make efforts to improve the rank minimization performance while retaining computation efficiency by exploring the use of a non-convex surrogate of the rank function, namely the so-called Schatten quasi p norm (0 < p < 1). Although the resulting optimization problem is non-convex, we show, for the first time, that a first-order critical point can be approximated to arbitrary accuracy in polynomial time using the proposed potential reduction algorithm in this dissertation. / We then apply the resulting potential reduction algorithm to Sensor Network Localization problem. Currently, a popular approach to localization is using convex relaxation. In a typical application of this approach, the localization problem is first formulated as a rank-constrained semidefinite program (SDP), where the rank corresponds to the target dimension in which the sensors should be localized. Then, the non-convex rank constraint is either dropped or replaced by a convex approximation, thus resulting into a convex optimization problem. Our potential reduction algorithm is applied to the localization problem while the Schatten quasi p norm is employed in localization aiming to minimize the solution rank. Moreover, we show that the first-order critical point, the output of the algorithm, is already suffcient for recovering the node locations in the target dimension if the input instance satisfies certain conditions which has been shown in the literature. Finally, our simulation results show that in many scenarios, our proposed algorithm can achieve better localization in terms of accuracy than the popular SDP relaxations of the problem and its variations. / 线性约束下的矩阵秩最小化在诸多领域有着广泛的应用。然而由于秩的复杂性质，许多研究致力于寻找它的替代函数。例如，核模是其中非常流行的一种，研究已经表明其在理论和应用上的有效性。由于这些替代函数往往具有秩函数所没有的特性，比如凸性，可微分性等等，正是由于这些特性使得其在计算上有着更高的效率保证。在本文中，我们希望通过一种非凸的替代函数来提高这种替代方法的有效性，并且在同时希望能继续保持在计算上的效率，我们使用的这种非凸替代函数被称为Schatten 拟p 模（0 < p < 1）。尽管该优化问题的目标函数是非凸的，我们还是能够第一次证明可以在多项式时间内以任意精度逼近该问题的一阶临界点。 / 同时，我们将得到的势削减算法应用于传感器网络定位问题中。当前，处理该问题的流行方法是使用凸放松。在这种方法中，我们首先将传感器网络定位问题写成一个秩限制条件下的半正定规划问题，它的秩限制条件对应于传感器所在空间的维度。在凸放松方法中，通常这些秩限制条件被直接去除，或者被一个凸函数取代。在我们的方法中，我们使用Schatten拟模作为惩罚函数来优化所得解的秩。我们还证明了在某些条件被满足的情况下，该问题的一阶临界点已经足够用来还原节点的位置。最后我们对多种情景做了模拟实验，结果表明我们提出的算法在准确性上相比流行的半正定规划及其衍生的方法更有优势。 / Ji, Senshan. / Thesis Ph.D. Chinese University of Hong Kong 2015. / Includes bibliographical references (leaves 102-110). / Abstracts also in Chinese. / Title from PDF title page (viewed on 06, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

Leung, Kin Pang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 70-71). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Model --- p.4 / Chapter 2.1 --- The Basic Thurstonian Model --- p.4 / Chapter 2.2 --- The Thurstonian Model with Mean Structure in 3-object Ranking Data --- p.6 / Chapter 3 --- Implementation Using the Mx --- p.13 / Chapter 4 --- Simulation Study --- p.21 / Chapter 4.1 --- 2 covariate values --- p.23 / Chapter 4.2 --- 4 covariate values --- p.23 / Chapter 4.3 --- 10 covariate values --- p.23 / Chapter 4.4 --- 50 covariate values --- p.24 / Chapter 5 --- Discussion --- p.37 / Chapter A --- Sample Mx script-2 covariate values --- p.39 / Chapter B --- Sample Mx script-50 covariate values --- p.60 / Bibliography --- p.70

Includes bibliographical references (p. 38-40).

This thesis consists of two topics: statistical selection and profile monitoring. Statistical selection is related to ranking and selection in simulation and profile monitoring is related to statistical process control. Ranking and selection (R&S) is to select a system with the largest or smallest performance measure among a finite number of simulated alternatives with some guarantee about correctness. Fully sequential procedures have been shown to be efficient, but their actual probabilities of correct selection tend to be higher than the nominal level, implying that they consume unnecessary observations. In the first part, we study three conservativeness sources in fully sequential indifference-zone (IZ) procedures and use experiments to quantify the impact of each source in terms of the number of observations, followed by an asymptotic analysis on the impact of the critical one. Then we propose new asymptotically valid procedures that lessen the critical conservativeness source, by mean update with or without variance update. Experimental results showed that new procedures achieved meaningful improvement on the efficiency. The second part is developing a wavelet-based distribution-free tabular CUSUM chart based on adaptive thresholding. WDFTCa is designed for rapidly detecting shifts in the mean of a high-dimensional profile whose noise components have a continuous nonsingular multivariate distribution. First computing a discrete wavelet transform of the noise vectors for randomly sampled Phase I (in-control) profiles, WDFTCa uses a matrix-regularization method to estimate the covariance matrix of the wavelet-transformed noise vectors; then those vectors are aggregated (batched) so that the nonoverlapping batch means of the wavelet-transformed noise vectors have manageable covariances. Lower and upper in-control thresholds are computed for the resulting batch means of the wavelet-transformed noise vectors using the associated marginal Cornish-Fisher expansions that have been suitably adjusted for between-component correlations. From the thresholded batch means of the wavelet-transformed noise vectors, Hotelling’s T^2-type statistics are computed to set the parameters of a CUSUM procedure. To monitor shifts in the mean profile during Phase II (regular) operation, WDFTCa computes a similar Hotelling’s T^2-type statistic from successive thresholded batch means of the wavelet-transformed noise vectors using the in-control thresholds; then WDFTCa applies the CUSUM procedure to the resulting T^2-type statistics. Experimentation with several normal and nonnormal test processes revealed that WDFTCa outperformed existing nonadaptive profile-monitoring schemes.

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Statistical methods based on nonparametric and distribution-free procedures require the use of order statistics. Order statistics are also used in many parametric estimation and testing problems. With the introduction of modern high speed computers, order statistics have gained more importance in recent years in statistical inference - the main reason being that ranking a large number of observations manually was difficult and time consuming in the past, which is no longer the case at present because of the availability of high speed computers. Also, applications of order statistics require in many cases the use of numerical tables and computer is needed to construct these tables.In this thesis, some basic concepts and results involving order statistics are provided. Typically, application of the Theory of Permanents in the distribution of order statistics are discussed. Further, the correlation coefficient between the smallest observation (Y1) and the largest observation (Y,,) of a random sample of size n from two gamma populations, where (n-1) observations of the sample are from one population and the remaining observation is from the other population, is presented. / Department of Mathematical Sciences

Thesis (Ph. D.)--University of Iowa, 2007. / Supervisor: W. Nick Street. Includes bibliographical references (leaves 83-89).

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