Os modelos multiníveis são uma classe de modelos úteis na análise de bases de dados com estrutura hierárquica. No presente trabalho propõem-se os modelos multiníveis com resposta Weibull, nos quais são considerados interceptos aleatórios na modelagem dos dois parâmetros da distribuição da variável resposta. Os modelos aqui propostos são flexíveis devido a que a distribuição dos interceptos aleatórios pode der escolhida entre uma das seguintes quatro distribuições: normal, log--gama, logística e Cauchy. Uma extensão dos modelos é apresentada na qual é possível incluir na parte sistemática dos dois parâmetros da distribuição da variável resposta interceptos e inclinações aleatórias com distribuição normal bivariada. A estimação dos parâmetros é realizada pelo método de máxima verossimilhança usando a quadratura de Gauss--Hermite para aproximar a função de verossimilhança. Um pacote em linguagem R foi desenvolvido especialmente para a estimação dos parâmetros, predição dos efeitos aleatórios e para a obtenção dos resíduos nos modelos propostos. Adicionalmente, por meio de um estudo de simulação foi avaliado o impacto nas estimativas dos parâmetros do modelo ao assumir incorretamente a distribuição dos interceptos aleatórios. / Multilevel models are a class of models useful in the analysis of datasets with hierarchical structure. In the present work we propose multilevel Weibull models in which random intercepts are considered to model the two parameters of the Weibull distribution. The proposed models are flexible due to random intercepts distribution can be chosen from one of the four following distributions: normal, log-gamma, logistics and Cauchy. An extension of the models is presented in which we can include, in the systematic part of the two parameters of the distribution, random intercepts and slopes with a bivariate normal distribution. The parameter estimation is performed by maximum likelihood method using the Gauss Hermite quadrature to approximate the likelihood function. A package in R language was especially developed to obtain parameter estimation, random effects predictions and residuals for the proposed models. Additionally, through a simulation study we investigated the misspecification random effect distribution on estimated parameter for the proposed model

Lo, Kwok Yuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 52-53). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Scope of the thesis --- p.3 / Chapter 2 --- Lifetime Model --- p.4 / Chapter 2.1 --- Introduction --- p.4 / Chapter 2.2 --- Weibull Distribution --- p.4 / Chapter 2.3 --- Step-Stress Experiment --- p.5 / Chapter 3 --- Maximum Likelihood Estimation of Model Parameters --- p.9 / Chapter 3.1 --- Introduction --- p.9 / Chapter 3.2 --- Maximum Likelihood Estimation --- p.10 / Chapter 3.3 --- Fisher Information Matrix --- p.13 / Chapter 3.4 --- Numerical Methods improving Newton's method. --- p.17 / Chapter 3.4.1 --- Initial values --- p.18 / Chapter 3.4.2 --- Fisher-Scoring method --- p.19 / Chapter 4 --- Optimal Experimental Design --- p.21 / Chapter 4.1 --- Introduction --- p.21 / Chapter 4.2 --- Optimal Criteria --- p.22 / Chapter 4.3 --- Optimal Stress-changing-time Proportion --- p.23 / Chapter 4.3.1 --- Optimal n versus the shape parameter B --- p.24 / Chapter 4.3.2 --- "Optimal n versus the parameters ao, a1" --- p.27 / Chapter 4.3.3 --- Optimal n versus the initial stress level x1 --- p.32 / Chapter 4.3.4 --- Optimal n versus the censoring time t2 --- p.33 / Chapter 4.4 --- Sensitivity Analysis --- p.34 / Chapter 4.4.1 --- Effects of the shape parameter B --- p.34 / Chapter 4.4.2 --- "Effects of the parameters ao, al" --- p.37 / Chapter 5 --- Conclusion Remarks and Further Research --- p.39 / Chapter A --- Simulation Algorithm for a Weibull Type-I Censored Simple Step-Stress Model --- p.41 / Chapter B --- Expected values of Fisher Information Matrix --- p.42 / Chapter C --- "Derivation of P(A1, A2)" --- p.50 / Bibliography --- p.52

To meet increasing competition, get products to market in the shortest possible time, and satisfy heightened customer expectations, products must be made more robust and fewer failures must be observed in a short development period. In this circumstance, assessing product reliability based on degradation data at high stress levels becomes necessary. This assessment is accomplished through accelerated degradation tests (ADT). These tests involve over stress testing in which instead of life product performance is measured as it degrades over time. Due to the role these tests play in determining proper reliability estimates for the product, it is necessary to scientifically design these test plans so as to save time and expense and provide more accurate estimates of reliability for a given number of test units and test time. In ADTs, several decision variables such as inspection frequency,the sample size, and the termination time at each stress level are important. In this research, an optimal plan is developed for the design of accelerated degradation test with a reciprocal Weibull degradation data using the mean time to failure (MTTF) as the minimizing criteria. A non linear integer programming problem is developed under the constraint that the total experimental cost does not exceed a pre-determined budget. The optimal combination of sample size, inspection frequency and the termination time at each stress level is found. A case example based on Light Emitting Diode (LED) example is used to illustrate the proposed method. Sensitivity analyses on the cost parameters and the parameters of the underlying probability distribution are performed to assess the robustness of the proposed method.

A surge function is a mathematical function of the form f(x)=axpe-bx. We simplify the surge function by holding p constant at 1 and investigate the simplified form as a potential model to represent the full peak of a stream discharge hydrograph. The previously studied Weibull and gamma distributions are included for comparison. We develop an analysis algorithm which produces the best-fit parameters for every peak for each model function, and we process the data with a MATLAB script that uses spectral analysis to filter year-long, 15-minute, stream-discharge data sets. The filtering is necessary to locate the concave-upward inflection points used to separate the data set into its constituent, individual peaks. The Levenberg-Marquardt algorithm is used to iteratively estimate the unknown parameters for each version of the modeled peak by minimizing the sum of squares of residuals. The results allow goodness-of-fit comparisons between the three model functions, as well as a comparison of peaks at the same gage through the year of record. Application of these methods to five rivers from three distinct hydrologic regions shows that the simple surge function is a special case of the gamma distribution, which is known to be useful as a modeling function for a full-peak hydrograph. The study also confirms that the Weibull distribution produces good fits to 15-minute hydrograph data.

This report presents the descriptive data analysis and failure time modeling that can be used to find out the characteristics and pattern of failure time. Descriptive data analysis includes the mean, median, 1st quartile, 3rd quartile, frequency, standard deviation, skewness, kurtosis, minimum, maximum and range. Models like exponential distribution, gamma distribution, normal distribution, lognormal distribution, Weibull distribution and log-logistic distribution have been studied for failure time data. The data in this report comes from the South Texas Project that was collected during the last 40 years. We generated more than 1000 groups for STP failure time data based on Mfg Part Number. In all, the top twelve groups of failure time data have been selected as the study group. For each group, we were able to perform different models and obtain the parameters. The significant level and p-value were gained by Kolmogorov-Smirnov test, which is a method of goodness of fit test that represents how well the distribution fits the data. The In this report, Weibull distribution has been proved as the most appropriate model for STP dataset. Among twelve groups, eight groups come from Weibull distribution. In general, Weibull distribution is powerful in failure time modeling. / text

Constant False Alarm Rate (CFAR) Processors are important for applications where thousands of detection tests are made per second, such as in radar. This thesis introduces a new method for CFAR threshold estimation that is particularly applicable to sound source detection with distributed microphone systems. The novel CFAR Processor exploits the near symmetry about 0 for the acoustic pixel values created by steered-response coherent power in conjunction with a partial whitening preprocessor to estimate thresholds for positive values, which represent potential targets. To remove the low frequency components responsible for degrading CFAR performance, fixed and adaptive high-pass filters are applied. A relation is proposed and it tested the minimum high-pass cut-off frequency and the microphone geometry. Experimental results for linear, perimeter and planar arrays illustrate that for desired false alarm (FA) probabilities ranging from 10-1 and 10-6, a good CFAR performance can be achieved by modeling the coherent power with Chi-square and Weibull distributions and the ratio of desired over experimental FA probabilities can be limited within an order of magnitude.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 105-108). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 105-108). Also available on the Internet.

Ce travail porte sur les tests d'adéquation en fiabilité, à la fois pour les systèmes non réparables et les systèmes réparables. Les tests d'adéquation sont des outils efficaces pour vérifier la pertinence d'un modèle pour un jeu de données. Pour les systèmes non réparables, la loi exponentielle et la loi de Weibull sont les lois de durée de vie les plus utilisées en fiabilité. Une comparaison exhaustive des tests d'adéquation pour la loi exponentielle est présentée pour des données complètes et censurées, suivie par des recommandations d'utilisation de ces tests. La loi de Weibull à deux paramètres permet de modéliser des taux de hasard décroissants et croissants contrairement à la loi exponentielle qui suppose un taux de hasard constant. Cependant, il existe moins de tests d'adéquation à la loi de Weibull dans la littérature. Une revue exhaustive des tests existant est effectuée et deux familles de tests exacts sont preésentées. La première famille est la famille des tests basés sur la vraisemblance et la deuxième est la famille des tests basés sur la transformée de Laplace. Des propriétés asymptotiques des nouvelles statistiques de tests sont établies. Une comparaison complète des tests d'adéquation pour la loi de Weibull est effectuée. Des recommandations sur les tests les plus puissants sont données en fonction des caractéristiques du jeu de donnés testé. Pour les systèmes réparables, de nouveaux tests d'adéquation sont développés pour des modèles de maintenance imparfaite avec à la fois des maintenances correctives et des maintenances préventives déterministes. Ces tests sont exacts et peuvent être appliqués à des petits jeux de données. Finalement, des applications à de vrais jeux de données issus de l'industrie sont effectuées pour des systèmes réparables et des systèmes non réparables. / This work deals with goodness-of-fit (GOF) tests in reliability for both non repairable and repairable systems. GOF tests are efficient techniques to check the relevance of a model for a given data set. For non repairable systems, the Exponential and Weibull distributions are the most used lifetimes distributions in reliability. A comprehensive comparison study of the GOF tests for the Exponential distribution is presented for complete and censored samples followed by recommendations about the use of the tests. The two-parameter Weibull distribution allows decreasing and increasing failure rates unlike the Exponential distribution that makes the assumption of a constant hazard rate. Yet, there exist less GOF tests in the literature for the Weibull distribution. A comprehensive review of the existing GOF tests is done and two new families of exact GOF tests are introduced. The first family is the likelihood based GOF tests and the second is the family of tests based on the Laplace transform. Theoretical asymptotic properties of some new tests statistics are established. A comprehensive comparison study of the GOF tests for the Weibull distribution is done. Recommendations about the most powerful tests are given depending on the characteristics of the tested data sets. For repairable systems, new GOF tests are developed for imperfect maintenance models when both corrective maintenance and deterministic preventive maintenance are performed. These tests are exact and can be applied to small data sets. Finally, illustrative applications to real data sets from industry are carried out for repairable and non repairable systems.

nÃo hà / Para determinar o potencial eÃlico de uma regiÃo à de fundamental importÃncia que seja realizado um estudo sobre as caracterÃsticas intrÃnsecas do vento do local. Este trabalho aborda alguns mÃtodos numÃricos a serem empregados no cÃlculo dos parÃmetros da distribuiÃÃo de Weibull que auxilia no estudo da velocidade do vento, para que haja uma correta definiÃÃo das condiÃÃes naturais existentes. O prÃvio conhecimento destas informaÃÃes coopera no processo de tomada de decisÃo sobre a viabilidade tÃcnica na instalaÃÃo de novos parques eÃlicos industriais. à realizada uma anÃlise estatÃstica entre sete sistemas matemÃticos conhecidos da literatura para estimar os parÃmetros ( k ) de forma e ( c ) de escala da curva de distribuiÃÃo de frequÃncias de Weibull. SÃo utilizados dados de velocidade do vento e de potÃncia eÃlica de duas cidades litorÃneas do Estado do Cearà pertencentes à regiÃo Nordeste do Brasil, Icapuà e Camocim. Os mÃtodos apurados no desenvolvimento desta pesquisa sÃo: MÃtodo GrÃfico, MÃtodo da MÃxima VerossimilhanÃa, MÃtodo da MÃxima VerossimilhanÃa Modificado, MÃtodo EmpÃrico, MÃtodo do Momento, MÃtodo da Energia PadrÃo e MÃtodo da Energia Equivalente. A realizaÃÃo da anÃlise comparativa de eficiÃncia e exatidÃo entre estes, compreende a aplicaÃÃo dos seguintes testes estatÃsticos: AnÃlise de VariÃncia (R2 ) , Raiz Quadrada dos Erros QuadrÃticos MÃdios (RMSE) e Teste do Qui-quadrado (X 2 ) . / To determine the wind potential of a region is of paramount importance that a study be conducted on the intrinsic characteristics of the wind site. This paper address some numerical methods to be used in calculating the parameters of the Weibull distribution the aids in the study of wind speed, so there is a correct definition of natural conditions existing. The previous knowledge of this information assist in the decision-making process on the technical feasibility of installing industrial wind farms. It perform a statistical analysis of seven mathematical systems known in the literature for estimating the parameters ( k ) form and ( c ) scale of the frequency distribution curve of Weibull. Data are used for wind speed and wind power from two coastal cities of Cearà State belonging to Northeast Brazil, Icapuà and Camocim. The methods to be employed in the development of this research are: Graphical Method, Maximum Likelihood Method, Maximum Likelihood Modified Method, Empirical Method, Moment Method, Energy Pattern Factor Method and the Equivalent Energy Method. The realization of the comparative analysis of efficiency and accuracy among these include the application of the following statistical tests: ANOVA(R2 ) , Square Root of Average Quadratic Errors (RMSE) and Chi-square (X 2 ).