Let (X, Y) be bivariate normal random vectors which represent the responses as a result of Treatment 1 and Treatment 2. The statistical inference about the bivariate normal distribution parameters involving missing data with both treatment samples is considered. Assuming the correlation coefficient ρ of the bivariate population is known, the MLE of population means and variance (ξ, η, and σ2) are obtained. Inferences about these parameters are presented. Procedures of constructing confidence interval for the difference of population means ξ – η and testing hypothesis about ξ – η are established. The performances of the new estimators and testing procedure are compared numerically with the method proposed in Looney and Jones (2003) on the basis of extensive Monte Carlo simulation. Simulation studies indicate that the testing power of the method proposed in this thesis study is higher.

In statistical analysis, distributional assumptions are needed to apply parametric procedures. Assumptions about underlying distribution should be true for accurate statistical inferences. Goodness-of-fit tests are used for checking the validity of the distributional assumptions. To apply some of the goodness-of-fit tests, the unknown population parameters are estimated. The null distributions of test statistics become complicated or depend on the unknown parameters if population parameters are replaced by their estimators. This will restrict the use of the test. Goodness-of-fit statistics which are invariant to parameters can be used if the distribution under null hypothesis is a location-scale distribution. For location and scale invariant goodness-of-fit tests, there is no need to estimate the unknown population parameters. However, approximations are used in some of those tests. Different types of estimation and approximation techniques are used in this study to compute goodness-of-fit statistics for complete and censored samples from univariate distributions as well as complete samples from bivariate normal distribution. Simulated power properties of the goodness-of-fit tests against a broad range of skew and symmetric alternative distributions are examined to identify the estimation effects in goodness-of-fit tests. The main aim of this thesis is to modify goodness-of-fit tests by using different estimators or approximation techniques, and finally see the effect of estimation on the power of these tests.

Suppose two or more variables are jointly normally distributed. If there is a common relationship between these variables it would be very important to quantify this relationship by a parameter called the correlation coefficient which measures its strength, and the use of it can develop an equation for predicting, and ultimately draw testable conclusion about the parent population. This research focused on the correlation coefficient ρ for the bivariate and trivariate normal distribution when equal variances and equal covariances are considered. Particularly, we derived the maximum Likelihood Estimators (MLE) of the distribution parameters assuming all of them are unknown, and we studied the properties and asymptotic distribution of . Showing this asymptotic normality, we were able to construct confidence intervals of the correlation coefficient ρ and test hypothesis about ρ. With a series of simulations, the performance of our new estimators were studied and were compared with those estimators that already exist in the literature. The results indicated that the MLE has a better or similar performance than the others.

The survival of our nation, during a nuclear exchange, depends upon an effective national defense structure. The prime weapon system in this defense structure is the ballistic missile. Although many factors enter into an evaluation of the effectiveness of a ballistic missile, one of the most important measure is accuracy. Without an accurate weapon system we have no weapon system. The Department of Defense has places emphasis on using a method of accuracy evaluation called "Probably Circular Error (CEP)." Probably Circular Error is defined as "The radius of a circle, centered at the intended target, within which 50% of the missiles would be expected to impact" or "The probability is 0.50 that an individual missile will impact within a circle whose radius is equal to the CEP." The statistical techniques and assumptions used in generation a CEP value will be investigated.

The low grades that students score in some statistical units in Kenyan universities is of great concern and has evoked research interest in the teaching of some of the units and the students’ learning of the statistical content. The aim of the study was to investigate the difficulties undergraduate students experience in the learning of bivariate normal distribution in a Kenyan university. The research also aimed to answer the following research questions on the difficulties undergraduate students encounter in the learning of bivariate normal distribution. The first research question was based on the reasons why students find learning of bivariate normal distribution difficult and the second research question was to find the reasons why students experience such difficulties in learning bivariate normal distribution. The target population for this study included lecturers teaching statistics in the university, and second- and third- year students enrolled or who have previously completed the probability and statistics III unit, where the bivariate normal distribution content is covered. In selecting students for the study, the simple random sampling technique was employed while convenient sampling was used to select lecturers who participated in the study. A mixed methods design was adopted for this study where both quantitative and qualitative data was collected. A total of 175 students and six lecturers participated in this research study. All students who participated in the study did a bivariate normal distribution test (Appendix 1) designed by the researcher and then filled in a questionnaire (Appendix 2). The lecturers who participated in the study filled in an open-ended questionnaire (Appendix 3). The results showed that undergraduate students have difficulties in learning bivariate normal distribution. This is because most of them could neither state the bivariate normal distribution nor solve any of the application questions on the content. The students find it difficult to learn and comprehend the bivariate normal distribution equation with its many parameters and constants of the two random independent variables. The results also showed that students could not state the normal distribution equation nor could they solve questions on the normal distribution, which forms the foundational knowledge required for effective learning of the bivariate normal distribution content. ii Based on the results, the study recommended that emphasis should be placed on the basic and foundational knowledge of the normal distribution content and its applications before teaching bivariate normal distribution in probability and statistics III. In addition, it is recommended that all students should be involved in the learning of basic content to enable them to understand all parameters and constants in the equations and their applications. The study also recommends that lecturers revise the foundational knowledge and content related to the bivariate normal distribution before introducing and teaching the bivariate normal distribution content. This study also recommends that the university should consider a change of curriculum by teaching the bivariate normal distribution, as an introductory course to the unit under the multivariate distributions in statistics, in third year of the students’ studies. ; ; / Mathematics Education / M. Sc. (Mathematics Education)

Stacy (1962)首先提出廣義伽瑪分配 (generalized gamma distribution)，此分布被廣泛應用於存活分析 (survival analysis) 以及可靠度 (reliability) 中壽命時間的資料描述。事實上，像是指數分配 (exponential distribution)、韋伯分配 (Weibull distribution) 以及伽瑪分配 (gamma distribution) 都是廣義伽瑪分配的一個特例。 Bologna (1987)提出一個特殊的雙變量廣義伽瑪分配 (bivariate generalized gamma distribution) 可以經由雙變量常態分配 (bivariate normal distribution) 所推得。我們根據他的想法，提出多變量廣義伽瑪分配可以經由多變量常態分配所推得。在過去的研究中，學者們做了許多有關雙變量伽瑪分配。當我們提到雙變量常態分配，由於其分配的型式為唯一的，所以沒人任何人對其分配的型式有疑問。然而，雙變量伽瑪分配卻有很多不同的型式。 在這篇論文中的架構如下。在第二章中，我們介紹並討論雙變量廣義伽瑪分配可以經由雙變量常態分配所推得，接著推導參數估計以及介紹模擬的程序。在第三章中，我們介紹一些對稱以及非對稱的雙變量伽瑪分配，接著拓展到雙變量廣義伽瑪分配，有關參數的估計以及模擬結果也將在此章中討論。在第三章最後，我們建構參數的敏感度分析 (sensitivity analysis)。最後，在第四章中，我們陳述結論以及未來研究方向。 / The generalized gamma distribution was introduced by Stacy (1962). This distribution is useful to describe lifetime data when conducting survival analysis and reliability. In fact, it includes the widely used exponential, Weibull, and gamma distributions as special cases. Bologna (1987) showed that a special bivariate genenralized gamma distribution can be derived from a bivariate normal distribution. Follow his idea, we show that a multivariate generalized gamma distribution can be derived from a multivariate normal distribution. In the past, researchers spend much time in working on a bivariate gamma distribution. When a bivariate normal distribution is mentioned, no one feels puzzled about its form, since it has only one form. However, there are various forms of bivariate gamma distributions. In this paper is as following. In Chapter 2, we introduce and discuss the bivariate generalized gamma distribution, then the multivariate generalized gamma distribution is derived. We also develop parameters estimation and simulation procedure. In Chapter 3, we introduce some symmetrical and asymmetrical bivariate gamma distributions, then they are extended to the bivariate generalized gamma distributions. Problems of parameters estimation and simulation results are also discussed in Chapter 3. Besides, sensitivity analyses of parameters estimation are conducted. Finally, we state conclusion and future work in Chapter 4.

Made available in DSpace on 2016-08-17T14:52:40Z (GMT). No. of bitstreams: 1 Dissertacao-AntonioAdolphoMartinsRaposo.pdf: 6219934 bytes, checksum: 92f0e1fb7c3d703fcf27aae305b549f2 (MD5) Previous issue date: 2016-02-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / To plan and operate properly a Smart Grid (SG), many new technical considerations in the context of distribution systems, must be considered, for example: stability (due to installation of Distributed Generation (DG), the load and generation dispatch, management of energy storage devices and the assessment of the impact of electric vehicle connection on the distribution system. The main prerequisite for many of these new functions in the distribution system control center is to determine the electrical network state (magnitude and angle of nodal voltages) in real time from measurement devices installed in it. In the transmission system control centers, this task is performed by the state estimation tool. Thus, the Distribution System State Estimation (DSSE) is one of the cornerstones for the implementation of a SG. The presence of a small number of measurements can make the grid unobservable in the context of the DSSE. That is, the state variables (magnitude and angle of the node voltages of all bus) can not be determined from a set of measurements by a state estimator. Due to this, it is usually added a large number of pseudo measurements to the existing measurement plan to ensure observability and to enable the DSSE. A drawback with this strategy is that the accuracy of the estimated state is compromised due to the fact that the errors associated with the pseudo measurements are considerably higher than those relating to real measurements. Consequently, it is necessary to allocate meters (voltage magnitude, active and reactive power flows, current magnitudes, etc.) to guarantee the accuracy of the DSEE. The meter placement problem for the state estimation in the transmission networks is usually carried out with the objective of assuring the observability. On the other hand, the meter placement for the EERD aims to minimize probabilistic index associated with the errors between the true and estimated state vectors. An important component of the method used to solve the meters placement problem is a probabilistic technique used to estimate the objective function. Due to the nonlinear nature of DSSE problem, the best option has been to use the Monte Carlo Simulation (MCS). A disadvantage of the MCS to estimate the objective function of the allocation problem is its high computational cost due to the need to solve a nonlinear state estimation problem for each sample element. The main objective of this dissertation is to propose a probabilistic techniques to improve the computational performance of existing methodologies for meter placement without reducing the accuracy of the estimated ix state. This compromise has been established using two strategies. In the first one, a linear model is used to estimate the state and the MCS is applied to determine the risks of the objective function. In the second one, a closed analytical formula is used to determine the risks based on the linearized model. Furthermore, the improved versions of the meter placement algorithms proposed in this dissertation consider the effect of the correlation among the measurements. The proposed meter placement algorithms were tested in the British distribution system of 95 bus. The tests results demonstrate that the introduction of the proposed strategies in a meter placement algorithm significantly reduced its computational cost. Moreover, it can be observed that there were improvements in accuracy in some cases, because the risk estimates provided by MCS are not accurate with small samples. / Para planejar e operar adequadamente uma Rede Elétrica Inteligente (REI), muitas novas considerações técnicas, no âmbito de sistemas de distribuição, devem ser apreciadas, por exemplo: a estabilidade devido a instalação de Geração Distribuída (GD), o despacho de carga e geração, o gerenciamento de dispositivos de armazenamento de energia e a avaliação do impacto da conexão de veículos elétricos na rede de distribuição. O principal pré-requisito para muitas destas novas funções do centro de controle do sistema de distribuição é a determinação do estado da rede elétrica (módulo e a fase das tensões nodais) em tempo real a partir de dispositivos de medição nela instalados. Em centros de controle de sistemas de transmissão esta tarefa é realizada por ferramentas de estimação de estado. Desta forma, a Estimação de Estado em Redes de Distribuição (EERD) é um dos alicerces para a implantação de uma REI. A presença de um número reduzido de medições pode tornar a rede elétrica não observável no âmbito da EERD. Isto é, as variáveis de estado (módulo e fase das tensões nodais em todas as barras) não podem ser determinadas a partir de um conjunto de medições por um estimador de estado. Devido a isto, geralmente adiciona-se um grande número de pseudo-medições ao plano de medição existente para assegurar a observabilidade e viabilizar a EERD. Um problema com esta estratégia é que a precisão do estado estimado é comprometida devido ao fato de que os erros associados com as pseudo-medições são consideravelmente maiores do que aqueles referentes às medições reais. Consequentemente é necessário alocar medidores (magnitude das tensões, fluxos de potência ativa e reativa, magnitude das correntes, etc.) para garantir a precisão do EERD. O problema de alocação de medidores para a estimação de estado em redes de transmissão é, geralmente, realizado com o objetivo de assegurar a observabilidade. Por outro lado, a alocação de medidores para EERD é realizada visando minimizar índices probabilísticos associados com os erros entre os vetores de estado estimado e verdadeiro. Um componente importante do método usado para resolver o problema de alocação de medidores é a técnica probabilística usada para estimar a função objetivo. Devido à natureza não-linear do problema de EERD, a melhor opção tem sido utilizar a Simulação Monte Carlo (SMC). Uma desvantagem da SMC para estimar a função objetivo do problema de alocação é o seu alto custo computacional devido a necessidade de resolver um problema de estimação de estado não-linear para cada vii elemento da amostra. O principal objetivo desta dissertação é propor técnicas probabilísticas para melhorar o desempenho computacional de metodologias existentes para alocação de medidores sem sacrificar a precisão do estado estimado. Este compromisso foi estabelecido usando-se duas estratégias. Na primeira, um modelo linearizado é usado para estimar o estado e a SMC para determinar os riscos da função objetivo. Na segunda, uma fórmula analítica fechada é usada para determinar os riscos com base no modelo linearizado. Além disso, as versões melhoradas dos algoritmos de alocação propostos nesta dissertação consideram o efeito da correlação entre as medições. As metodologias de alocação propostas foram testadas no sistema de distribuição britânico de 95 barras. Os resultados dos testes demonstraram que a introdução das estratégias propostas em um algoritmo de alocação de medidores reduziu significativamente o seu custo computacional. Além disso, pode-se observar que ocorreram melhorias na precisão em alguns casos, pois as estimativas dos riscos fornecidas pela SMC não são precisas com pequenas amostras.

In or around 2010, a nationwide reevaluation of the allowable properties for southern pine dimension lumber was initiated. This led to a 2013 reduction in the design values of visually graded southern pine dimension lumber and a resulting decrease in its commercial and utility value. This change compelled researchers and industry professionals to ponder what could be done to shore up the value of solid-sawn southern pine products going forward and potentially increase design values if appropriate. In pursuit of this question, this dissertation looks closely at three areas: 1) the possibility this reduction in mechanical performance is not merely limited to southern pine structural lumber but can also be observed in other solid-sawn softwood products and species, 2) flaws that might exist in commonly utilized statistical models for estimating allowable properties in lumber, and 3) the feasibility of using existing technologies to begin to compensate for the economic and/or utility losses attributed to the recent reduction in design values. This work is comprised of an introduction, a conclusion, and three independent content chapters utilizing a variety of statistical techniques to investigate whether strength and stiffness reduction might also be occurring in southern pine (and Douglasir) utility crossarms, evaluate the propriety of using a Weibull distribution model for estimating allowable properties in dimension lumber, and gauge the suitability of nondestructive testing methods for potentially identifying high-value premium grades in solid-sawn softwood products.