Researchers conduct life testing on objects of interest in an attempt to determine their life distribution as a means of studying their reliability (or survivability). Determining the life distribution of the objects under study helps manufacturers to identify potential faults, and to improve quality. Researchers sometimes conduct accelerated life tests (ALTs) to ensure that failure among the tested units is earlier than what could result under normal operating (or environmental) conditions. Moreover, such experiments allow the experimenters to examine the effects of high levels of one or more stress factors on the lifetimes of experimental units. Examples of stress factors include, but not limited to, cycling rate, dosage, humidity, load, pressure, temperature, vibration, voltage, etc. A special class of ALT is step-stress accelerated life testing. In this type of experiments, the study sample is tested at initial stresses for a given period of time. Afterwards, the levels of the stress factors are increased in agreement with prefixed points of time called stress-change times. In practice, time and resources are limited; thus, any experiment is expected to be constrained to a deadline which is called a termination time. Hence, the observed information may be subjected to Type-I censoring. This study discusses maximum likelihood inferential methods for the parameters of multiple step-stress models from a generalized Birnbaum-Saunders distribution under time constraint alongside other inference-related problems. A couple of general inference frameworks are studied; namely, the observed likelihood (OL) framework, and the expectation-maximization (EM) framework. The last-mentioned framework is considered since there is a possibility that Type-I censored data are obtained. In the first framework, the scoring algorithm is used to get the maximum likelihood estimators (MLEs) for the model parameters. In the second framework, EM-based algorithms are utilized to determine the required MLEs. Obtaining observed information matrices under both frameworks is also discussed. Accordingly, asymptotic and bootstrap-based interval estimators for the model parameters are derived. Model discrimination within the considered generalized Birnbaum-Saunders distribution is carried out by likelihood ratio test as well as by information-based criteria. The discussed step-stress models are illustrated by analyzing three real-life datasets. Accordingly, establishing optimal multiple step-stress test plans based on cost considerations and three optimality criteria is discussed. Since maximum likelihood estimators are obtained by numerical optimization that involves maximizing some objective functions, optimization methods used, and their software implementations in R are discussed. Because of the computational aspects are in focus in this study, the benefits of parallel computing in R, as a high-performance computational approach, are briefly addressed. Numerical examples and Monte Carlo simulations are used to illustrate and to evaluate the methods presented in this thesis. / Thesis / Doctor of Science (PhD)

Rieck & Nedelman (1991) propuseram um modelo de regressão log-linear tendo como base a distribuição Birnbaum-Saunders (Birnbaum & Saunders, 1969a). O modelo proposto pelos autores vem sendo bastante explorado e tem se mostrado uma ótima alternativa a outros modelos propostos na literatura, como por exemplo, os modelos de regressão Weibull, gama e lognormal. No entanto, até o presente momento, não existe nenhum estudo tratando de refinamentos para as estatísticas da razão de verossimilhanças e escore nesta classe de modelos de regressão. Assim, um dos objetivos desta tese é obter um fator de correção de Bartlett para a estatística da razão de verossimilhanças e um fator de correção tipo-Bartlett para a estatística escore nesse modelo. Estes ajustes melhoram a aproximação da distribuição nula destas estatísticas pela distribuição qui-quadrado de referência. Adicionalmente, objetiva-se obter ajustes para a estatística da razão de verossimilhanças sinalizada. Tais ajustes melhoram a aproximação desta estatística pela distribuição normal padrão. Recentemente, uma nova estatística de teste foi proposta por Terrell (2002), a qual o autor denomina estatística gradiente. Esta estatística foi derivada a partir da estatística escore e da estatística de Wald modificada (Hayakawa & Puri, 1985). A combinação daquelas duas estatísticas resulta em uma estatística muito simples de ser calculada, não envolvendo, por exemplo, nenhum cálculo matricial como produto e inversa de matrizes. Esta estatística foi recentemente citada por Rao (2005): \"The suggestion by Terrell is attractive as it is simple to compute. It would be of interest to investigate the performance of the [gradient] statistic.\" Caminhando na direção da sugestão de Rao, outro objetivo da tese é obter uma expansão assintótica para a distribuição da estatística gradiente sob uma sequência de alternativas de Pitman convergindo para a hipótese nula a uma taxa de convergência de n^{-1/2} utilizando a metodologia desenvolvida por Peers (1971) e Hayakawa (1975). Em particular, mostramos que, até ordem n^{-1/2}, a estatística gradiente segue distribuição qui-quadrado central sob a hipótese nula e distribuição qui-quadrado não central sob a hipótese alternativa. Também temos como objetivo comparar o poder local deste teste com o poder local dos testes da razão de verossimilhanças, de Wald e escore. Finalmente, aplicaremos a expansão assintótica derivada na tese em algumas classes particulares de modelos. / The Birnbaum-Saunders regression model is commonly used in reliability studies.We address the issue of performing inference in this class of models when the number of observations is small. Our simulation results suggest that the likelihood ratio and score tests tend to be liberal when the sample size is small. We derive Bartlett and Bartlett-type correction factors which reduce the size distortion of the tests. Additionally, we also consider modified signed log-likelihood ratio statistics in this class of models. Finally, the asymptotic expansion of the distribution of the gradient test statistic is derived for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate n^{-1/2}, n being the sample size. Comparisons of the local powers of the gradient, likelihood ratio, Wald and score tests reveal no uniform superiority property.

The aim of this work is to present an inference and diagnostic study of an extension of the lifetime distribution family proposed by Birnbaum and Saunders (1969a,b). This extension is obtained by considering a skew-elliptical distribution instead of the normal distribution. Specifically, in this work we develop a Birnbaum-Saunders (BS) distribution type based on scale mixtures of skew-normal distributions (SMSN). The resulting family of lifetime distributions represents a robust extension of the usual BS distribution. Based on this family, we reproduce the usual properties of the BS distribution, and present an estimation method based on the EM algorithm. In addition, we present regression models associated with the BS distributions (based on scale mixtures of skew-normal), which are developed as an extension of the sinh-normal distribution (Rieck and Nedelman, 1991). For this model we consider an estimation and diagnostic study for uncensored data. / O objetivo deste trabalho é apresentar um estudo de inferência e diagnóstico em uma extensão da família de distribuições de tempos de vida proposta por Birnbaum e Saunders (1969a,b). Esta extensão é obtida ao considerar uma distribuição skew-elíptica em lugar da distribuição normal. Especificamente, neste trabalho desenvolveremos um tipo de distribuição Birnbaum-Saunders (BS) baseda nas distribuições mistura de escala skew-normal (MESN). Esta família resultante de distribuições de tempos de vida representa uma extensão robusta da distribuição BS usual. Baseado nesta família, vamos reproduzir as propriedades usuais da distribuição BS, e apresentar um método de estimação baseado no algoritmo EM. Além disso, vamos apresentar modelos de regressão associado à distribuições BS (baseada na distribuição mistura de escala skew-normal), que é desenvolvida como uma extensão da distribuição senh-normal (Rieck e Nedelman, 1991), para estes vamos considerar um estudo de estimação e diagnóstisco para dados sem censura.

The aim of this work is to present an inference and diagnostic study of an extension of the lifetime distribution family proposed by Birnbaum and Saunders (1969a,b). This extension is obtained by considering a skew-elliptical distribution instead of the normal distribution. Specifically, in this work we develop a Birnbaum-Saunders (BS) distribution type based on scale mixtures of skew-normal distributions (SMSN). The resulting family of lifetime distributions represents a robust extension of the usual BS distribution. Based on this family, we reproduce the usual properties of the BS distribution, and present an estimation method based on the EM algorithm. In addition, we present regression models associated with the BS distributions (based on scale mixtures of skew-normal), which are developed as an extension of the sinh-normal distribution (Rieck and Nedelman, 1991). For this model we consider an estimation and diagnostic study for uncensored data. / O objetivo deste trabalho é apresentar um estudo de inferência e diagnóstico em uma extensão da família de distribuições de tempos de vida proposta por Birnbaum e Saunders (1969a,b). Esta extensão é obtida ao considerar uma distribuição skew-elíptica em lugar da distribuição normal. Especificamente, neste trabalho desenvolveremos um tipo de distribuição Birnbaum-Saunders (BS) baseda nas distribuições mistura de escala skew-normal (MESN). Esta família resultante de distribuições de tempos de vida representa uma extensão robusta da distribuição BS usual. Baseado nesta família, vamos reproduzir as propriedades usuais da distribuição BS, e apresentar um método de estimação baseado no algoritmo EM. Além disso, vamos apresentar modelos de regressão associado à distribuições BS (baseada na distribuição mistura de escala skew-normal), que é desenvolvida como uma extensão da distribuição senh-normal (Rieck e Nedelman, 1991), para estes vamos considerar um estudo de estimação e diagnóstisco para dados sem censura.

Submitted by Rafael Santana (rafael.silvasantana@ufpe.br) on 2017-04-26T17:07:37Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Carolina Marchant.pdf: 5792192 bytes, checksum: adbd82c79b286d2fe2470b7955e6a9ed (MD5) / Made available in DSpace on 2017-04-26T17:07:38Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Carolina Marchant.pdf: 5792192 bytes, checksum: adbd82c79b286d2fe2470b7955e6a9ed (MD5) Previous issue date: 2016-10-31 / CAPES; BOLSA DO CHILE. / In the last decades, univariate Birnbaum-Saunders models have received considerable attention in the literature. These models have been widely studied and applied to fatigue, but they have also been applied to other areas of the knowledge. In such areas, it is often necessary to model several variables simultaneously. If these variables are correlated, individual analyses for each variable can lead to erroneous results. Multivariate regression models are a useful tool of the multivariate analysis, which takes into account the correlation between variables. In addition, diagnostic analysis is an important aspect to be considered in the statistical modeling. Furthermore, multivariate quality control charts are powerful and simple visual tools to determine whether a multivariate process is in control or out of control. A multivariate control chart shows how several variables jointly affect a process. First, we propose, derive and characterize multivariate generalized logarithmic Birnbaum-Saunders distributions. Also, we propose new multivariate generalized Birnbaum-Saunders regression models. We use the method of maximum likelihood estimation to estimate their parameters through the expectation-maximization algorithm. We carry out a simulation study to evaluate the performance of the corresponding estimators based on the Monte Carlo method. We validate the proposed models with a regression analysis of real-world multivariate fatigue data. Second, we conduct a diagnostic analysis for multivariate generalized Birnbaum-Saunders regression models. We consider the Mahalanobis distance as a global influence measure to detect multivariate outliers and use it for evaluating the adequacy of the distributional assumption. Moreover, we consider the local influence method and study how a perturbation may impact on the estimation of model parameters. We implement the obtained results in the R software, which are illustrated with real-world multivariate biomaterials data. Third and finally, we develop a robust methodology based on multivariate quality control charts for generalized Birnbaum-Saunders distributions with the Hotelling statistic. We use the parametric bootstrap method to obtain the distribution of this statistic. A Monte Carlo simulation study is conducted to evaluate the proposed methodology, which reports its performance to provide earlier alerts of out-of-control conditions. An illustration with air quality real-world data of Santiago-Chile is provided. This illustration shows that the proposed methodology can be useful for alerting episodes of extreme air pollution. / Nas últimas décadas, o modelo Birnbaum-Saunders univariado recebeu considerável atenção na literatura. Esse modelo tem sido amplamente estudado e aplicado inicialmente à modelagem de fadiga de materiais. Com o passar dos anos surgiram trabalhos com aplicações em outras áreas do conhecimento. Em muitas das aplicações é necessário modelar diversas variáveis simultaneamente incorporando a correlação entre elas. Os modelos de regressão multivariados são uma ferramenta útil de análise multivariada, que leva em conta a correlação entre as variáveis de resposta. A análise de diagnóstico é um aspecto importante a ser considerado no modelo estatístico e verifica as suposições adotadas como também sua sensibilidade. Além disso, os gráficos de controle de qualidade multivariados são ferramentas visuais eficientes e simples para determinar se um processo multivariado está ou não fora de controle. Este gráfico mostra como diversas variáveis afetam conjuntamente um processo. Primeiro, propomos, derivamos e caracterizamos as distribuições Birnbaum-Saunders generalizadas logarítmicas multivariadas. Em seguida, propomos um modelo de regressão Birnbaum-Saunders generalizado multivariado. Métodos para estimação dos parâmetros do modelo, tal como o método de máxima verossimilhança baseado no algoritmo EM, foram desenvolvidos. Estudos de simulação de Monte Carlo foram realizados para avaliar o desempenho dos estimadores propostos. Segundo, realizamos uma análise de diagnóstico para modelos de regressão Birnbaum-Saunders generalizados multivariados. Consideramos a distância de Mahalanobis como medida de influência global de detecção de outliers multivariados utilizando-a para avaliar a adequacidade do modelo. Além disso, desenvolvemos medidas de diagnósticos baseadas em influência local sob alguns esquemas de perturbações. Implementamos a metodologia apresentada no software R, e ilustramos com dados reais multivariados de biomateriais. Terceiro, e finalmente, desenvolvemos uma metodologia robusta baseada em gráficos de controle de qualidade multivariados para a distribuição Birnbaum-Saunders generalizada usando a estatística de Hotelling. Baseado no método bootstrap paramétrico encontramos aproximações da distribuição desta estatística e obtivemos limites de controle para o gráfico proposto. Realizamos um estudo de simulação de Monte Carlo para avaliar a metodologia proposta indicando seu bom desempenho para fornecer alertas precoces de processos fora de controle. Uma ilustração com dados reais de qualidade do ar de Santiago-Chile é fornecida. Essa ilustração mostra que a metodologia proposta pode ser útil para alertar sobre episódios de poluição extrema do ar, evitando efeitos adversos na saúde humana.

Rieck & Nedelman (1991) propuseram um modelo de regressão log-linear tendo como base a distribuição Birnbaum-Saunders (Birnbaum & Saunders, 1969a). O modelo proposto pelos autores vem sendo bastante explorado e tem se mostrado uma ótima alternativa a outros modelos propostos na literatura, como por exemplo, os modelos de regressão Weibull, gama e lognormal. No entanto, até o presente momento, não existe nenhum estudo tratando de refinamentos para as estatísticas da razão de verossimilhanças e escore nesta classe de modelos de regressão. Assim, um dos objetivos desta tese é obter um fator de correção de Bartlett para a estatística da razão de verossimilhanças e um fator de correção tipo-Bartlett para a estatística escore nesse modelo. Estes ajustes melhoram a aproximação da distribuição nula destas estatísticas pela distribuição qui-quadrado de referência. Adicionalmente, objetiva-se obter ajustes para a estatística da razão de verossimilhanças sinalizada. Tais ajustes melhoram a aproximação desta estatística pela distribuição normal padrão. Recentemente, uma nova estatística de teste foi proposta por Terrell (2002), a qual o autor denomina estatística gradiente. Esta estatística foi derivada a partir da estatística escore e da estatística de Wald modificada (Hayakawa & Puri, 1985). A combinação daquelas duas estatísticas resulta em uma estatística muito simples de ser calculada, não envolvendo, por exemplo, nenhum cálculo matricial como produto e inversa de matrizes. Esta estatística foi recentemente citada por Rao (2005): \"The suggestion by Terrell is attractive as it is simple to compute. It would be of interest to investigate the performance of the [gradient] statistic.\" Caminhando na direção da sugestão de Rao, outro objetivo da tese é obter uma expansão assintótica para a distribuição da estatística gradiente sob uma sequência de alternativas de Pitman convergindo para a hipótese nula a uma taxa de convergência de n^{-1/2} utilizando a metodologia desenvolvida por Peers (1971) e Hayakawa (1975). Em particular, mostramos que, até ordem n^{-1/2}, a estatística gradiente segue distribuição qui-quadrado central sob a hipótese nula e distribuição qui-quadrado não central sob a hipótese alternativa. Também temos como objetivo comparar o poder local deste teste com o poder local dos testes da razão de verossimilhanças, de Wald e escore. Finalmente, aplicaremos a expansão assintótica derivada na tese em algumas classes particulares de modelos. / The Birnbaum-Saunders regression model is commonly used in reliability studies.We address the issue of performing inference in this class of models when the number of observations is small. Our simulation results suggest that the likelihood ratio and score tests tend to be liberal when the sample size is small. We derive Bartlett and Bartlett-type correction factors which reduce the size distortion of the tests. Additionally, we also consider modified signed log-likelihood ratio statistics in this class of models. Finally, the asymptotic expansion of the distribution of the gradient test statistic is derived for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate n^{-1/2}, n being the sample size. Comparisons of the local powers of the gradient, likelihood ratio, Wald and score tests reveal no uniform superiority property.

Survival analytic methods help to analyze lifetime data arising from medical and reliability experiments. The popular proportional hazards model, proposed by Cox (1972), is widely used in survival analysis to study the effect of risk factors on lifetimes. An important assumption in regression type analysis is that all relative risk factors should be included in the model. However, not all relative risk factors are observed due to measurement difficulty, inaccessibility, cost considerations, and so on. These unobservable risk factors can be modelled by the so-called frailty model, originally introduced by Vaupel et al. (1979). Furthermore, the frailty model is also applicable to clustered data. Cluster data possesses the feature that observations within the same cluster share similar conditions and environment, which are sometimes difficult to observe. For example, patients from the same family share similar genetics, and patients treated in the same hospital share the same group of profes- sionals and same environmental conditions. These factors are indeed hard to quantify or measure. In addition, this type of similarity introduces correlation among subjects within clusters. In this thesis, a semi-parametric frailty model is proposed to address aforementioned issues. The baseline hazards function is approximated by a piecewise constant function and the frailty distribution is assumed to be a Birnbaum-Saunders distribution. Due to the advancement in modern medical sciences, many diseases are curable, which in turn leads to the need of incorporating cure proportion in the survival model. The frailty model discussed here is further extended to a mixture cure rate frailty model by integrating the frailty model into the mixture cure rate model proposed originally by Boag (1949) and Berkson and Gage (1952). By linking the covariates to the cure proportion through logistic/logit link function and linking observable covariates and unobservable covariates to the lifetime of the uncured population through the frailty model, we obtain a flexible model to study the effect of risk factors on lifetimes. The mixture cure frailty model can be reduced to a mixture cure model if the effect of frailty term is negligible (i.e., the variance of the frailty distribution is close to 0). On the other hand, it also reduces to the usual frailty model if the cure proportion is 0. Therefore, we can use a likelihood ratio test to test whether the reduced model is adequate to model the given data. We assume the baseline hazard to be that of Weibull distribution since Weibull distribution possesses increasing, constant or decreasing hazard rate, and the frailty distribution to be Birnbaum-Saunders distribution. D ́ıaz-Garc ́ıa and Leiva-Sa ́nchez (2005) proposed a new family of life distributions, called generalized Birnbaum-Saunders distribution, including Birnbaum-Saunders distribution as a special case. It allows for various degrees of kurtosis and skewness, and also permits unimodality as well as bimodality. Therefore, integration of a generalized Birnbaum-Saunders distribution as the frailty distribution in the mixture cure frailty model results in a very flexible model. For this general model, parameter estimation is carried out using a marginal likelihood approach. One of the difficulties in the parameter estimation is that the likelihood function is intractable. The current technology in computation enables us to develop a numerical method through Monte Carlo simulation, and in this approach, the likelihood function is approximated by the Monte Carlo method and the maximum likelihood estimates and standard errors of the model parameters are then obtained numerically by maximizing this approximate likelihood function. An EM algorithm is also developed for the Birnbaum-Saunders mixture cure frailty model. The performance of this estimation method is then assessed by simulation studies for each proposed model. Model discriminations is also performed between the Birnbaum-Saunders frailty model and the generalized Birnbaum-Saunders mixture cure frailty model. Some illustrative real life examples are presented to illustrate the models and inferential methods developed here. / Thesis / Doctor of Science (PhD)

The modern supply chain is long, complex, interconnected and global, and plays a fundamental role in business competitiveness. These conditions, along with various supply chain management trends in recent years have increased risks in supply chains which threaten supply chain performance. Greater impact, especially on cost, from an increased threat of supply disruptions is one area of particular concern. Companies today are struggling to find effective means to manage this increased risk and avoid adverse financial impacts. An approach to managing supply disruption risk in supply chains based on the minimization of the total cost of ownership (TCO) of the supply chain is explored in this thesis. Insights are provided into an appropriate view of supply chain risk and a general four step risk management process to guide the design and evaluation of a new risk management tool based on such an approach. A prototype of the new total cost-based, modeling and simulation tool was created in partnership with ProModel Corporation and a government contractor that requested to remain anonymous. A preliminary assessment of the effectiveness of this tool in minimizing TCO and providing an interface useable by non-modelers is provided. This study also reviews and compares a sample set of current supply chain risk management methods and tools and compares them with the new tool for relevance in aiding users in managing supply disruption risk. Based on literature findings and preliminary feedback from pilot contextual demonstrations of the tool, the total cost approach to risk modeling appears promising, although the execution needs to be improved with further enhancements made to the prototype tool. In this preliminary study and evaluation, sufficient evidence is not available to determine that the new prototype tool is any more effective than other currently available risk management tools to provide necessary information to make supply chain risk management decisions that minimize TCO of a supply chain. Suggestions for further development of the tool, especially for improvement of the total cost approach, are provided as well as a preliminary evaluation procedure and survey instruments for a more robust evaluation of the new tool.

Cette thèse étudie d'un point de vue historique la genèse de la cohomologie des groupes, théorie qui vit le jour dans les années 1940. Il s'agit d'une théorie à la fois algébrique, au sens où elle donne des résultats sur les groupes, et topologique par les méthodes qu'elle met en œuvre . Le présent travail analyse les mécanismes par lesquels la topologie et l'algèbre se sont interpénétrées pour donner naissance à cette théorie abstraite et élaborée, en mettant notamment en perspective ce phénomène par rapport à ceux, plus globaux, de la naissance et de l'expansion de l'algèbre moderne. Y sont notamment discutées l'influence d'Emmy Noether dans l'algébrisation de la topologie et les motivations respectives de Heinz Hopf et d'Eilenberg & Mac Lane les ayant menés à l'élaboration de l'homologie des groupes. L'analyse minutieuse de plusieurs articles phares - dus aux auteurs cités précédemment mais aussi à Schur, Vietoris ou encore Eckmann - permet de mettre en lumière le fait que la volonté de répondre à des problèmes mathématiques précis fut peut-être plus motrice, dans l'émergence de cette théorie architectonique qu'est la cohomologie des groupes, que de grandes idées directrices conçues au sein de représentations structurales des mathématiques.

This dissertation discusses the potential of the snare drum as a solo concert instrument. Four pieces from a collection entitled The Noble Snare are used for demonstration ("Homily" by Milton Babbitt, "Composed Improvisation for Snare Drum" by John Cage, "Peeping Tom" by Dan Senn, and "The Noble Snare" by Stuart Saunders Smith). In the absence of many traditional musical devices (i.e. melody and harmony), alternative means of expression are used by the composer. Each piece is discussed with regard to its distinctive compositional approach and inherent performance issues. Information is also given pertaining to the background of the Noble Snare series. This includes: the inspiration for the project, editorial issues, and its influence on snare drum performance. Much of this research was completed through interviews by with author with Sylvia Smith, publisher of The Noble Snare and owner of Smith Publications.