<p>This thesis investigates nonparametric inference under multiple independent samples with various modes of censoring, and also presents results concerning Pitman Closeness under Progressive Type-II right censoring. For the nonparametric inference with multiple independent samples, the case of Type-II right censoring is first considered. Two extensions to this are then discussed: doubly Type-II censoring, and Progressive Type-II right censoring. We consider confidence intervals for quantiles, prediction intervals for order statistics from a future sample, and tolerance intervals for a population proportion. Benefits of using multiple samples over one sample are discussed. For each of these scenarios, we consider simulation as an alternative to exact calculations. In each case we illustrate the results with data from the literature. Furthermore, we consider two problems concerning Pitman Closeness and Progressive Type-II right censoring. We derive simple explicit formulae for the Pitman Closeness probabilities of the order statistics to population quantiles. Various tables are given to illustrate these results. We then use the Pitman Closeness measure as a criterion for determining the optimal censoring scheme for samples drawn from the exponential distribution. A general result is conjectured, and demonstrated in special cases</p> / Doctor of Philosophy (PhD)

Continuous-time multi-state models are widely used in modeling longitudinal data of disease processes with multiple transient states, yet the analysis is complex when subjects are observed periodically, resulting in interval censored data. Recently, most studies focused on modeling the true disease progression as a discrete time stationary Markov chain, and only a few studies have been carried out regarding non-homogenous multi-state models in the presence of interval-censored data. In this dissertation, several likelihood-based methodologies were proposed to deal with interval censored data in multi-state models. Firstly, a continuous time version of a homogenous Markov multi-state model with backward transitions was proposed to handle uneven follow-up assessments or skipped visits, resulting in the interval censored data. Simulations were used to compare the performance of the proposed model with the traditional discrete time stationary Markov chain under different types of observation schemes. We applied these two methods to the well-known Nun study, a longitudinal study of 672 participants aged ≥ 75 years at baseline and followed longitudinally with up to ten cognitive assessments per participant. Secondly, we constructed a non-homogenous Markov model for this type of panel data. The baseline intensity was assumed to be Weibull distributed to accommodate the non-homogenous property. The proportional hazards method was used to incorporate risk factors into the transition intensities. Simulation studies showed that the Weibull assumption does not affect the accuracy of the parameter estimates for the risk factors. We applied our model to data from the BRAiNS study, a longitudinal cohort of 531 subjects each cognitively intact at baseline. Last, we presented a parametric method of fitting semi-Markov models based on Weibull transition intensities with interval censored cognitive data with death as a competing risk. We relaxed the Markov assumption and took interval censoring into account by integrating out all possible unobserved transitions. The proposed model also allowed for incorporating time-dependent covariates. We provided a goodness-of-fit assessment for the proposed model by the means of prevalence counts. To illustrate the methods, we applied our model to the BRAiNS study.

Consideramos um modelo de regressão linear simples, em que tanto a variável resposta como a independente estão sujeitas a censura intervalar. Como motivação utilizamos um estudo em que o objetivo é avaliar a possibilidade de previsão dos resultados de um exame audiológico comportamental a partir dos resultados de um exame audiológico eletrofisiológico. Calculamos intervalos de previsão para a variável resposta, analisamos o comportamento dos estimadores de máxima verossimilhança obtidos sob o modelo proposto e comparamos seu desempenho com aquele de estimadores obtidos de um modelo de regressão linear simples usual, no qual a censura dos dados é desconsiderada. / We consider a simple linear regression model in which both variables are interval censored. To motivate the problem we use data from an audiometric study designed to evaluate the possibility of prediction of behavioral thresholds from physiological thresholds. We develop prediction intervals for the response variable, obtain the maximum likelihood estimators of the proposed model and compare their performance with that of estimators obtained under ordinary linear regression models.

Nesta tese, apresenta-se um modelo de confiabilidade estatística para aplicação em dados de vida de um produto, buscando classificar três modos de falhas distintos associados à ocorrência de falhas prematuras, aleatórias e por desgaste. A ocorrência dos três modos de falhas segue os princípios de aplicação dos modelos teóricos por riscos concorrentes e seccionais. O modelo proposto utiliza duas distribuições de Weibull, com dois e três parâmetros, e uma distribuição exponencial. A distribuição de Weibull com dois parâmetros tem por objetivo representar os modos de falhas prematuras: a distribuição de Weibull com três parâmetros busca capturar os modos de falhas por desgaste; a distribuição exponencial mede a ocorrência de falhas aleatórias decorrentes de uso operacional de um produto. Considera-se que falhas prematuras e por desgaste ocorram seqüencialmente, enquanto falhas aleatórias ocorram de forma concorrente às falhas prematuras e por desgaste tão logo o produto seja colocado em operação. Para dimensionar o número de ocorrências vinculadas aos três modos de falhas são utilizados dados coletados durante o período de garantia e pós-garantia. Os dados de garantia são registros históricos do produtor e os dados da pós-garantia referem-se a informações obtidas de especialistas, já que dados após a garantia apresentam elevado nível de censura. Equações de confiabilidade e estimadores de máxima verossimilhança são apresentados para definir o perfil e os parâmetros do modelo proposto. Um estudo de caso com dados coletados de um equipamento elétrico-eletrônico subsidia a aplicação do modelo enquanto que um teste estatístico de ajuste de dados é utilizado para validar o referido modelo. / This thesis presents a reliability model for product life data presenting three different failure modes, associated with early, random and wear-out failures. The model is based on theoretical concepts related to competing risk and sectional models. The proposed model is structured based on two Weibull distributions, with two and three parameters, and one exponential distribution. The Weibull distribution with two parameters is aimed at modeling early failure modes; the Weibull distribution with three parameters models wear-out failure modes; the exponential distribution models random failures due to operational use. It is considered that early and wearout failures take place one after the other while random failures occur at the same time as early and wear-out failures as soon as the product starts operating. To measure each period related to the three failure modes, data from warranty and post-warranty periods are used. Warranty data are historical records; post-warranty data are gathered from experts, and are aimed at decreasing the degree of censoring in the data. Once the model is defined, reliability figures and maximum likelihood estimators are derived. Real data obtained from warranty claims on electricelectronic equipments are used to illustrate the developments proposed and a goodness-of-fit test is used to validate the performance of this model.

A model is proposed to provide inventory and maintenance guidance for a system of operating parts. This model is capable of handling a system with multiple operating components, unknown part lifetime failure distribution, and separately maintained parts. In this model, part reliability characteristics are used along with system costs to predict the required stocking levels and part replacement times. Two maintenance strategies are presented that have the unique characteristic of allowing flexible scheduling of replacements. A case study is completed comparing developed stocking policies to an existing policy. An estimation selection method is introduced and fit into the model for computing Weibull distribution parameters when part reliability is not well known. An algorithm is displayed that describes the implementation of the system model and data from practical case scenarios are conducted using this algorithm.

Consumer expenditure on food away from home in the United States has grown substantially in recent decades. Changes in the food service system, increased complexity of family structure, and the food policies made by government agencies have continued to influence the marketing, distribution, retailing, and demand for food products and the food industry. This study explores consumption behavior on food away from home (FAFH) and determines the differentiated impacts of economic and demographic variables on FAFH by type of meal and by type of facility among different household types. Each of the two systems of expenditures is estimated with two alternative econometric procedures to accommodate censoring in the dependent variables: the trivariate Tobit estimator and the multivariate sample selection estimator. Data for this study come from the 2008 and 2009 Consumer Expenditure Surveys, the most recent U.S. national household expenditure surveys conducted by the Bureau of Labor Statistics. Joint statistical significance of error correlations among equations justifies estimation of the sample selection systems. The opposite marginal effects on probabilities and expenditure levels of some variables highlight the advantage of the sample selection system over the Tobit system. Segmentation of the sample by household types is also justified with formal statistical tests. The empirical results indicate that the effects of demographic and socioeconomic factors on FAFH consumption vary by type of meal and by type of facility. Income, work hours, race, education, geographical region, and household composition are important factors. Food stamps have no impact on FAFH for married couples without children and single parenthood has conflicting effects on probabilities and conditional levels of expenditures.

The accelerated lifetime model is considered. To test the influence of the covariate we transform the model in a regression model. Since censoring is allowed this approach leads to a goodness-of-fit problem for regression functions under censoring. So nonparametric estimation of regression functions under censoring is investigated, a limit theorem for a L2-distance is stated and a test procedure is formulated. Finally a Monte Carlo procedure is proposed.

Cross-sectional surveys are often used in epidemiological studies to identify subjects with a disease. When estimating the survival function from onset of disease, this sampling mechanism introduces bias, which must be accounted for. If the onset times of the disease are assumed to be coming from a stationary Poisson process, this bias, which is caused by the sampling of prevalent rather than incident cases, is termed length-bias. A one-sample Kolomogorov-Smirnov type of goodness-of-fit test for right-censored length-biased data is proposed and investigated with Weibull, log-normal and log-logistic models. Algorithms detailing how to efficiently generate right-censored length-biased survival data of these parametric forms are given. Simulation is employed to assess the effects of sample size and censoring on the power of the test. Finally, the test is used to evaluate the goodness-of-fit using length-biased survival data of patients with dementia from the Canadian Study of Health and Aging.

Cross-sectional surveys are often used in epidemiological studies to identify subjects with a disease. When estimating the survival function from onset of disease, this sampling mechanism introduces bias, which must be accounted for. If the onset times of the disease are assumed to be coming from a stationary Poisson process, this bias, which is caused by the sampling of prevalent rather than incident cases, is termed length-bias. A one-sample Kolomogorov-Smirnov type of goodness-of-fit test for right-censored length-biased data is proposed and investigated with Weibull, log-normal and log-logistic models. Algorithms detailing how to efficiently generate right-censored length-biased survival data of these parametric forms are given. Simulation is employed to assess the effects of sample size and censoring on the power of the test. Finally, the test is used to evaluate the goodness-of-fit using length-biased survival data of patients with dementia from the Canadian Study of Health and Aging.

Being able to explore a relationship between two life events is of great interest to scientists from different disciplines. Some issues of particular concern are, for example, the connection between smoking cessation and pregnancy (Thompson and Pantoja-Galicia 2003), the interrelation between entry into marriage for individuals in a consensual union and first pregnancy (Blossfeld and Mills 2003), and the association between job loss and divorce (Charles and Stephens 2004, Huang 2003 and Yeung and Hofferth 1998). Establishing causation in observational studies is seldom possible. Nevertheless, if one of two events tends to precede the other closely in time, a causal interpretation of an association between these events can be more plausible. The role of longitudinal surveys is crucial, then, since they allow sequences of events for individuals to be observed. Thompson and Pantoja-Galicia (2003) discuss in this context several notions of temporal association and ordering, and propose an approach to investigate a possible relationship between two lifetime events. In longitudinal surveys individuals might be asked questions of particular interest about two specific lifetime events. Therefore the joint distribution might be advantageous for answering questions of particular importance. In follow-up studies, however, it is possible that interval censored data may arise due to several reasons. For example, actual dates of events might not have been recorded, or are missing, for a subset of (or all) the sampled population, and can be established only to within specified intervals. Along with the notions of temporal association and ordering, Thompson and Pantoja-Galicia (2003) also discuss the concept of one type of event "triggering" another. In addition they outline the construction of tests for these temporal relationships. The aim of this thesis is to implement some of these notions using interval censored data from longitudinal complex surveys. Therefore, we present some proposed tools that may be used for this purpose. This dissertation is divided in five chapters, the first chapter presents a notion of a temporal relationship along with a formal nonparametric test. The mechanisms of right censoring, interval censoring and left truncation are also overviewed. Issues on complex surveys designs are discussed at the end of this chapter. For the remaining chapters of the thesis, we note that the corresponding formal nonparametric test requires estimation of a joint density, therefore in the second chapter a nonparametric approach for bivariate density estimation with interval censored survey data is provided. The third chapter is devoted to model shorter term triggering using complex survey bivariate data. The semiparametric models in Chapter 3 consider both noncensoring and interval censoring situations. The fourth chapter presents some applications using data from the National Population Health Survey and the Survey of Labour and Income Dynamics from Statistics Canada. An overall discussion is included in the fifth chapter and topics for future research are also addressed in this last chapter.