In this dissertation, we apply Bayesian and Empirical Bayes methods for reliability growth models based on the power law process. We also apply Bayes methods for the study of microarrays, in particular, in the selection of differentially expressed genes. The power law process has been used extensively in reliability growth models. Chapter 1 reviews some basic concepts in reliability growth models. Chapter 2 shows classical inferences on the power law process. We also assess the goodness of fit of a power law process for a reliability growth model. In chapter 3 we develop Bayesian procedures for the power law process with failure truncated data, using non-informative priors for the scale and location parameters. In addition to obtaining the posterior density of parameters of the power law process, prediction inferences for the expected number of failures in some time interval and the probability of future failure times are also discussed. The prediction results for the software reliability model are illustrated. We compare our result with the result of Bar-Lev,S.K. et al. Also, posterior densities of several parametric functions are given. Chapter 4 provides Empirical Bayes for the power law process with natural conjugate priors and nonparametric priors. For the natural conjugate priors, two-hyperparameter prior and a more generalized three-hyperparameter prior are used. In chapter 5, we review some basic statistical procedures that are involved in microarray analysis. We will also present and compare several transformation and normalization methods for probe level data. The objective of chapter 6 is to select differentially expressed genes from tens of thousands of genes. Both classical methods (fold change, T-test, Wilcoxon Rank-sum Test, SAM and local Z-score and Empirical Bayes methods (EBarrays and LIMMA) are applied to obtain the results. Outputs of a typical classical method and a typical Empirical Bayes Method are discussed in detail.

Many software reliability growth models have beenanalyzed for measuring the growth of software reliability. In this dissertation, regression methods are explored to study software reliability models. First, two parametric linear models are proposed and analyzed, the simple linear regression and transformed linearregression corresponding to a power law process. Some software failure data sets do not follow the linear pattern. Analysis of popular real life data showed that these contain outliers andleverage values. Linear regression methods based on least squares are sensitive to outliers and leverage values. Even though the parametric regression methods give good results in terms of error measurement criteria, these results may not be accurate due to violation of the parametric assumptions. To overcome these difficulties, nonparametric regression methods based on ranks are proposed as alternative techniques to build software reliability models. In particular, monotone regre ssion and rank regression methods are used to evaluate the predictive capability of the models. These models are applied to real life data sets from various projects as well as to diverse simulated data sets. Both the monotone and the rank regression methods are robust procedures that are less sensitive to outliers and leverage values. In particular, the regression approach explains predictive properties of the mean time to failure for modeling the patterns of software failure times.In order to decide on model preference and to asses predictive accuracy of the mean time between failure time estimates for the defined data sets, the following error measurements evaluative criteria are used: the mean square error, mean absolute value difference, mean magnitude of relative error, mean magnitude oferror relative to the estimate, median of the absolute residuals, and a measure of dispersion. The methods proposed in this dissertation, when applied to real software failure data, give lesserror in terms of all the measurement criteria compared to other popular methods from literature. Experimental results show that theregression approach offers a very promising technique in software reliability growth modeling and prediction.

System testing is very time-consuming and costly, especially for complex high-cost and high-reliability systems. For this reason, the number of failures needed for the developmental phase of system testing should be relatively small in general. To assess the reliability growth of a repairable system, the generalized confidence interval and the modified signed log-likelihood ratio test for the scale parameter of the power-law process are studied concerning incomplete failure data. Specifically, some recorded failure times in the early developmental phase of system testing cannot be observed; this circumstance is essential to establish a warranty period or determine a maintenance phase for repairable systems. For the proposed generalized confidence interval, we have found that this method is not biased estimates which can be seen from the coverage probabilities obtained from this method being close to the nominal level 0.95 for all levels of γ and β. When the performance of the proposed method and the existing method are compared and validated regarding average widths, the simulation results show that the proposed method is superior to another method due to shorter average widths when the predetermined number of failures is small. For the proposed modified signed log-likelihood ratio test, we have found that this test performs well in controlling type I errors for complete failure data, and it has desirable powers for all parameters configurations even for the small number of failures. For incomplete failure data, the proposed modified signed log-likelihood ratio test is preferable to the signed log-likelihood ratio test in most situations in terms of controlling type I errors. Moreover, the proposed test also performs well when the missing ratio is up to 30% and n > 10. In terms of empirical powers, the proposed modified signed log-likelihood ratio test is superior to another test for most situations. In conclusion, it is quite clear that the proposed methods, the generalized confidence interval, and the modified signed log-likelihood ratio test, are practically useful to save business costs and time during the developmental phase of system testing since the only small number of failures is required to test systems, and it yields precise results.

In this thesis, the main objective is to study certain aspects of modeling failure time data of repairable systems under a competing risks framework. We consider two different models and propose more efficient Bayesian methods for estimating the parameters. In the first model, we discuss inferential procedures based on an objective Bayesian approach for analyzing failures from a single repairable system under independent competing risks. We examined the scenario where a minimal repair is performed at each failure, thereby resulting in that each failure mode appropriately follows a power-law intensity. Besides, it is proposed that the power-law intensity is reparametrized in terms of orthogonal parameters. Then, we derived two objective priors known as the Jeffreys prior and reference prior. Moreover, posterior distributions based on these priors will be obtained in order to find properties which may be optimal in the sense that, for some cases, we prove that these posterior distributions are proper and are also matching priors. In addition, in some cases, unbiased Bayesian estimators of simple closed-form expressions are derived. In the second model, we analyze data from multiple repairable systems under the presence of dependent competing risks. In order to model this dependence structure, we adopted the well-known shared frailty model. This model provides a suitable theoretical basis for generating dependence between the components failure times in the dependent competing risks model. It is known that the dependence effect in this scenario influences the estimates of the model parameters. Hence, under the assumption that the cause-specific intensities follow a PLP, we propose a frailty-induced dependence approach to incorporate the dependence among the cause-specific recurrent processes. Moreover, the misspecification of the frailty distribution may lead to errors when estimating the parameters of interest. Because of this, we considered a Bayesian nonparametric approach to model the frailty density in order to offer more flexibility and to provide consistent estimates for the PLP model, as well as insights about heterogeneity among the systems. Both simulation studies and real case studies are provided to illustrate the proposed approaches and demonstrate their validity. / Nesta tese, o objetivo principal é estudar certos aspectos da modelagem de dados de tempo de falha de sistemas reparáveis sob uma estrutura de riscos competitivos. Consideramos dois modelos diferentes e propomos métodos Bayesianos mais eficientes para estimar os parâmetros. No primeiro modelo, discutimos procedimentos inferenciais baseados em uma abordagem Bayesiana objetiva para analisar falhas de um único sistema reparável sob riscos competitivos independentes. Examinamos o cenário em que um reparo mínimo é realizado em cada falha, resultando em que cada modo de falha segue adequadamente uma intensidade de lei de potência. Além disso, propõe-se que a intensidade da lei de potência seja reparametrizada em termos de parâmetros ortogonais. Então, derivamos duas prioris objetivas conhecidas como priori de Jeffreys e priori de referência. Além disso, distribuições posteriores baseadas nessas prioris serão obtidas a fim de encontrar propriedades que podem ser ótimas no sentido de que, em alguns casos, provamos que essas distribuições posteriores são próprias e que também são matching priors. Além disso, em alguns casos, estimadores Bayesianos não-viesados de forma fechada são derivados. No segundo modelo, analisamos dados de múltiplos sistemas reparáveis sob a presença de riscos competitivos dependentes. Para modelar essa estrutura de dependência, adotamos o conhecido modelo de fragilidade compartilhada. Esse modelo fornece uma base teórica adequada para gerar dependência entre os tempos de falha dos componentes no modelo de riscos competitivos dependentes. Sabe-se que o efeito de dependência neste cenário influencia as estimativas dos parâmetros do modelo. Assim, sob o pressuposto de que as intensidades específicas de causa seguem um PLP, propomos uma abordagem de dependência induzida pela fragilidade para incorporar a dependência entre os processos recorrentes específicos da causa. Além disso, a especificação incorreta da distribuição de fragilidade pode levar a erros na estimativa dos parâmetros de interesse. Por isso, consideramos uma abordagem Bayesiana não paramétrica para modelar a densidade da fragilidade, a fim de oferecer mais flexibilidade e fornecer estimativas consistentes para o modelo PLP, bem como insights sobre a heterogeneidade entre os sistemas. São fornecidos estudos de simulação e estudos de casos reais para ilustrar as abordagens propostas e demonstrar sua validade.

Statistical analysis and modeling are useful for understanding the behavior of different phenomena. In this study we will focus on two areas of applications: Global warming and cancer research. Global Warming is one of the major environmental challenge people face nowadays and cancer is one of the major health problem that people need to solve. For Global Warming, we are interest to do research on two major contributable variables: Carbon dioxide (CO2) and atmosphere temperature. We will model carbon dioxide in the atmosphere data with a system of differential equations. We will develop a differential equation for each of six attributable variables that constitute CO2 in the atmosphere and a differential system of CO2 in the atmosphere. We are using real historical data on the subject phenomenon to develop the analytical form of the equations. We will evaluate the quality of the developed model by utilizing a retrofitting process. Having such an analytical system, we can obtain good estimates of the rate of change of CO2 in the atmosphere, individually and cumulatively as a function of time for near and far target times. Such information is quite useful in strategic planning of the subject matter. We will develop a statistical model taking into consideration all the attributable variables that have been identified and their corresponding response of the amount of CO2 in the atmosphere in the continental United States. The development of the statistical model that includes interactions and higher order entities, in addition to individual contributions to CO2 in the atmosphere, are included in the present study. The proposed model has been statistically evaluated and produces accurate predictions for a given set of the attributable variables. Furthermore, we rank the attributable variables with respect to their significant contribution to CO2 in the atmosphere. For Cancer Research, the object of the study is to probabilistically evaluate commonly used methods to perform survival analysis of medical patients. Our study includes evaluation of parametric, semi-parametric and nonparametric analysis of probability survival models. We will evaluate the popular Kaplan-Meier (KM), the Cox Proportional Hazard (Cox PH), and Kernel density (KD) models using both Monte Carlo simulation and using actual breast cancer data. The first part of the evaluation will be based on how these methods measure up to parametric analysis and the second part using actual cancer data. As expected, the parametric survival analysis when applicable gives the best results followed by the not commonly used nonparametric Kernel density approach for both evaluations using simulation and actual cancer data. We will develop a statistical model for breast cancer tumor size prediction for United States patients based on real uncensored data. When we simulate breast cancer tumor size, most of time these tumor sizes are randomly generated. We want to construct a statistical model to generate these tumor sizes as close as possible to the real patients' data given other related information. We accomplish the objective by developing a high quality statistical model that identifies the significant attributable variables and interactions. We rank these contributing entities according to their percentage contribution to breast cancer tumor growth. This proposed statistical model can also be used to conduct surface response analysis to identify the necessary restrictions on the significant attributable variables and their interactions to minimize the size of the breast tumor. We will utilize the Power Law process, also known as Non-homogenous Poisson Process and Weibull Process to evaluate the effectiveness of a given treatment for Stage I & II Ductal breast cancer patients. We utilize the shape parameter of the intensity function to evaluate the behavior of a given treatment with respect to its effectiveness. We will develop a differential equation that will characterize the behavior of the tumor as a function of time. Having such a differential equation, the solution of which once plotted will identify the rate of change of tumor size as a function of age. The structure of the differential equation consists of the significant attributable variables and their interactions to the growth of breast cancer tumor. Once we have developed the differential equations and its solution, we proceed to validate the quality of the proposed differential equations and its usefulness.

Um sistema reparável opera sob uma estratégia de manutenção que exige ações de recuperação preventiva em tempos pré-definidos e ações de reparo quando ocorre a perda de função do sistema. A manutenção preventiva (MP) é programada periodicamente e muitas vezes possui um intervalo de tempo fixo para ações. No entanto, as atividades de MP podem não restaurar o sistema para uma condição similar ao início de vida deste, mas para uma situação intermediária. Nesse caso, a MP é denominada de imperfeita. Além disso, ao longo da vida do sistema, são executados diferentes planos de manutenção com condições e atividades distintas que podem afetar a intensidade de falha de diferentes maneiras. Para modelar essas características da MP em um sistema reparável, propõe-se uma nova classe de modelo de fator de melhoria, denominado fator de melhoria variável que possibilita a modelagem da situação de manutenção perfeita. A formulação da função de verossimilhança foi desenvolvida para estimação dos parâmetros bem como desenvolvidos testes de verificação da qualidade de ajuste, intervalos de confiança para os parâmetros e otimização da periodicidade de realização da MP com base no enfoque dos novos modelos propostos. Os resultados foram aplicados em dados reais e verificou-se uma parametrização mais flexível a MP imperfeita e maior versatilidade nas análises de confiabilidade do sistema quando utilizado os novos modelos. / A repairable system operates under a maintenance strategy that calls for preventive repair actions at prescheduled times and the repair actions that restore system when failure occurs. The preventive maintenance (PM) is scheduled periodically and it often holds a fixed time interval for PM actions. However, PM activities are generally imperfect and cannot restore the system to as good as new condition but to an intermediate situation, which is called imperfect PM. In addition, throughout system life are implemented diverse maintenance policies with different activities and conditions that may affect the failure intensity in different ways. To model these PM characteristics, proposes a new model class of improvement factor called variable improvement factor that also enables modeling perfect maintenance situation. The likelihood function is developed for parameter estimation as well as goodness-of-fit tests and confidence intervals for the parameters are developed, and optimization of the PM intervals based on the proposed models is presented. The proposed model was applied to a data set and a more flexible parameterization for imperfect PM and greater versatility in the system reliability analysis were verified with the use of the new model.

Um sistema reparável opera sob uma estratégia de manutenção que exige ações de recuperação preventiva em tempos pré-definidos e ações de reparo quando ocorre a perda de função do sistema. A manutenção preventiva (MP) é programada periodicamente e muitas vezes possui um intervalo de tempo fixo para ações. No entanto, as atividades de MP podem não restaurar o sistema para uma condição similar ao início de vida deste, mas para uma situação intermediária. Nesse caso, a MP é denominada de imperfeita. Além disso, ao longo da vida do sistema, são executados diferentes planos de manutenção com condições e atividades distintas que podem afetar a intensidade de falha de diferentes maneiras. Para modelar essas características da MP em um sistema reparável, propõe-se uma nova classe de modelo de fator de melhoria, denominado fator de melhoria variável que possibilita a modelagem da situação de manutenção perfeita. A formulação da função de verossimilhança foi desenvolvida para estimação dos parâmetros bem como desenvolvidos testes de verificação da qualidade de ajuste, intervalos de confiança para os parâmetros e otimização da periodicidade de realização da MP com base no enfoque dos novos modelos propostos. Os resultados foram aplicados em dados reais e verificou-se uma parametrização mais flexível a MP imperfeita e maior versatilidade nas análises de confiabilidade do sistema quando utilizado os novos modelos. / A repairable system operates under a maintenance strategy that calls for preventive repair actions at prescheduled times and the repair actions that restore system when failure occurs. The preventive maintenance (PM) is scheduled periodically and it often holds a fixed time interval for PM actions. However, PM activities are generally imperfect and cannot restore the system to as good as new condition but to an intermediate situation, which is called imperfect PM. In addition, throughout system life are implemented diverse maintenance policies with different activities and conditions that may affect the failure intensity in different ways. To model these PM characteristics, proposes a new model class of improvement factor called variable improvement factor that also enables modeling perfect maintenance situation. The likelihood function is developed for parameter estimation as well as goodness-of-fit tests and confidence intervals for the parameters are developed, and optimization of the PM intervals based on the proposed models is presented. The proposed model was applied to a data set and a more flexible parameterization for imperfect PM and greater versatility in the system reliability analysis were verified with the use of the new model.

Les travaux présentés dans cette thèse concernent l'analyse statistique de cas particuliers du processus de Cox. Dans une première partie, nous proposons une synthèse des résultats existants sur le processus power-law (processus d'intensité puissance), synthèse qui ne peut être exhaustive étant donné la popularité de ce processus. Nous considérons une approche bayésienne pour l'inférence des paramètres de ce processus qui nous conduit à introduire et à étudier en détails une distribution que nous appelons loi H-B. Cette loi est une loi conjuguée. Nous proposons des stratégies d'élicitation des hyperparamètres et étudions le comportement des estimateurs de Bayes par des simulations. Dans un deuxième temps, nous étendons ces travaux au cas du processus d’intensité exponentielle (exponential-law process). De la même façon, nous définissons et étudions une loi conjuguée pour l'analyse bayésienne de ce dernier. Dans la dernière partie de la thèse, nous considérons un processus auto-excité qui intègre une covariable. Ce travail est motivé, à l'origine, par un problème de fiabilité qui concerne des données de défaillances de matériels exposés à des environnements sévères. Les résultats sont illustrés par des applications sur des données d'activités orageuses collectées dans deux départements français. Enfin, nous donnons quelques directions de travail et perspectives de futurs développements de l'ensemble de nos travaux. / The work presented in this PhD dissertation concerns the statistical analysis of some particular cases of the Cox process. In a first part, we study the power-law process (PLP). Since the literature for the PLP is abundant, we suggest a state-of-art for the process. We consider the classical approach and recall some important properties of the maximum likelihood estimators. Then we investigate a Bayesian approach with noninformative priors and conjugate priors considering different parametrizations and scenarios of prior guesses. That leads us to define a family of distributions that we name H-B distribution as the natural conjugate priors for the PLP. Bayesian analysis with the conjugate priors are conducted via a simulation study and an application on real data. In a second part, we study the exponential-law process (ELP). We review the maximum likelihood techniques. For Bayesian analysis of the ELP, we define conjugate priors: the modified- Gumbel distribution and Gamma-modified-Gumbel distribution. We conduct a simulation study to compare maximum likelihood estimates and Bayesian estimates. In the third part, we investigate self-exciting point processes and we integrate a power-law covariate model to this intensity of this process. A maximum likelihood procedure for the model is proposed and the Bayesian approach is suggested. Lastly, we present an application on thunderstorm data collected in two French regions. We consider a strategy to define a thunderstorm as a temporal process associated with the charges in a particular location. Some selected thunderstorms are analyzed. We propose a reduced maximum likelihood procedure to estimate the parameters of the Hawkes process. Then we fit some thunderstorms to the power-law covariate self-exciting point process taking into account the associated charges. In conclusion, we give some perspectives for further work.