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Doctor of Philosophy / Department of Statistics / Paul Nelson / Panels of judges are often used to estimate consumer preferences for m items such as food products. Judges can either evaluate each item on several ordinal scales and indirectly produce an overall ranking, or directly report a ranking of the items. A complete ranking orders all the items from best to worst. A partial ranking, as we use the term, only reports rankings of the best q out of m items. Direct ranking, the subject of this report, does not require the widespread but questionable practice of treating ordinal measurement as though they were on ratio or interval scales. Here, we develop and study segmentation models in which the panel may consist of relatively homogeneous subgroups, the segments. Judges within a subgroup will tend to agree among themselves and differ from judges in the other subgroups. We develop and study the statistical analysis of mixture models where it is not known to which segment a judge belongs or in some cases how many segments there are. Viewing segment membership indicator variables as latent data, an E-M algorithm was used to find the maximum likelihood estimators of the parameters specifying a mixture of Mallow’s (1957) distance models for complete and partial rankings. A simulation study was conducted to evaluate the behavior of the E-M algorithm in terms of such issues as the fraction of data sets for which the algorithm fails to converge and the sensitivity of initial values to the convergence rate and the performance of the maximum likelihood estimators in terms of bias and mean square error, where applicable. A Bayesian approach was developed and credible set estimators was constructed. Simulation was used to evaluate the performance of these credible sets as confidence sets. A method for predicting segment membership from covariates measured on a judge was derived using a logistic model applied to a mixture of Mallows probability distance models. The effects of covariates on segment membership were assessed. Likelihood sets for parameters specifying mixtures of Mallows distance models were constructed and explored.

This research has developed several models and methodologies with the aim of improving the accuracy and applicability of reliability predictions for complex repairable systems. A repairable system is usually defined as one that will be repaired to recover its functions after each failure. Physical assets such as machines, buildings, vehicles are often repairable. Optimal maintenance strategies require the prediction of the reliability of complex repairable systems accurately. Numerous models and methods have been developed for predicting system reliability. After an extensive literature review, several limitations in the existing research and needs for future research have been identified. These include the follows: the need for an effective method to predict the reliability of an asset with multiple preventive maintenance intervals during its entire life span; the need for considering interactions among failures of components in a system; and the need for an effective method for predicting reliability with sparse or zero failure data. In this research, the Split System Approach (SSA), an Analytical Model for Interactive Failures (AMIF), the Extended SSA (ESSA) and the Proportional Covariate Model (PCM), were developed by the candidate to meet the needs identified previously, in an effective manner. These new methodologies/models are expected to rectify the identified limitations of current models and significantly improve the accuracy of the reliability prediction of existing models for repairable systems. The characteristics of the reliability of a system will alter after regular preventive maintenance. This alternation makes prediction of the reliability of complex repairable systems difficult, especially when the prediction covers a number of imperfect preventive maintenance actions over multiple intervals during the asset's lifetime. The SSA uses a new concept to address this issue effectively and splits a system into repaired and unrepaired parts virtually. SSA has been used to analyse system reliability at the component level and to address different states of a repairable system after single or multiple preventive maintenance activities over multiple intervals. The results obtained from this investigation demonstrate that SSA has an excellent ability to support the making of optimal asset preventive maintenance decisions over its whole life. It is noted that SSA, like most existing models, is based on the assumption that failures are independent of each other. This assumption is often unrealistic in industrial circumstances and may lead to unacceptable prediction errors. To ensure the accuracy of reliability prediction, interactive failures were considered. The concept of interactive failure presented in this thesis is a new variant of the definition of failure. The candidate has made several original contributions such as introducing and defining related concepts and terminologies, developing a model to analyse interactive failures quantitatively and revealing that interactive failure can be either stable or unstable. The research results effectively assist in avoiding unstable interactive relationship in machinery during its design phase. This research on interactive failures pioneers a new area of reliability prediction and enables the estimation of failure probabilities more precisely. ESSA was developed through an integration of SSA and AMIF. ESSA is the first effective method to address the reliability prediction of systems with interactive failures and with multiple preventive maintenance actions over multiple intervals. It enhances the capability of SSA and AMIF. PCM was developed to further enhance the capability of the above methodologies/models. It addresses the issue of reliability prediction using both failure data and condition data. The philosophy and procedure of PCM are different from existing models such as the Proportional Hazard Model (PHM). PCM has been used successfully to investigate the hazard of gearboxes and truck engines. The candidate demonstrated that PCM had several unique features: 1) it automatically tracks the changing characteristics of the hazard of a system using symptom indicators; 2) it estimates the hazard of a system using symptom indicators without historical failure data; 3) it reduces the influence of fluctuations in condition monitoring data on hazard estimation. These newly developed methodologies/models have been verified using simulations, industrial case studies and laboratory experiments. The research outcomes of this research are expected to enrich the body of knowledge in reliability prediction through effectively addressing some limitations of existing models and exploring the area of interactive failures.

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.