In this thesis, we apply smoothed empirical likelihood method to investigate confidence intervals for the receiver operating characteristic (ROC) curve with right censoring. As a particular application of comparison of distributions from two populations, the ROC curve is constructed by the combination of cumulative distribution function and quantile function. Under mild conditions, the smoothed empirical likelihood ratio converges to chi-square distribution, which is the well-known Wilks's theorem. Furthermore, the performances of the empirical likelihood method are also illustrated by simulation studies in terms of coverage probability and average length of confidence intervals. Finally, a primary biliary cirrhosis data is used to illustrate the proposed empirical likelihood procedure.

In this thesis, we study two methods for inference of parameters in the accelerated failure time model with right censoring data. One is the Wald-type method, which involves parameter estimation. The other one is empirical likelihood method, which is based on the asymptotic distribution of likelihood ratio. We employ a monotone censored data version of Kendall estimating equation, and construct confidence intervals from both methods. In the simulation studies, we compare the empirical likelihood (EL) and the Wald-type procedure in terms of coverage accuracy and average length of confidence intervals. It is concluded that the empirical likelihood method has a better performance. We also compare the EL for Kendall’s rank regression estimator with the EL for other well known estimators and find advantages of the EL for Kendall estimator for small size sample. Finally, a real clinical trial data is used for the purpose of illustration.

Empirical likelihood (EL) is a recently developed nonparametric method of statistical inference. Owen’s 2001 book contains many important results for EL with uncensored data. However, fewer results are available for EL with right-censored data. In this dissertation, we first investigate a right-censored-data extension of Qin and Lawless (1994). They studied EL with uncensored data when the number of estimating equations is larger than the number of parameters (over-determined case). We obtain results similar to theirs for the maximum EL estimator and the EL ratio test, for the over-determined case, with right-censored data. We employ hazard-type constraints which are better able to handle right-censored data. Then we investigate EL with right-censored data and a k-sample mixed hazard-type constraint. We show that the EL ratio test statistic has a limiting chi-square distribution when k = 2. We also study the relationship between the constrained Kaplan-Meier estimator and the corresponding Nelson-Aalen estimator. We try to prove that they are asymptotically equivalent under certain conditions. Finally we present simulation studies and examples showing how to apply our theory and methodology with real data.

In this study, the most prominent goodness-of-fit tests for censored samples are reviewed. Power properties of goodness-of-fit statistics of the null hypothesis that a sample which is censored from right, left and both right and left which comes from uniform, normal and exponential distributions are investigated. Then, by a similar argument extreme value, student t with 6 degrees of freedom and generalized logistic distributions are discussed in detail through a comprehensive simulation study. A variety of real life applications are given. Suitable test statistics for testing the above distributions for censored samples are also suggested in the conclusion.

Doctor of Philosophy / Department of Statistics / Christopher Vahl / Veterinary drugs are used extensively for disease prevention and treatment in food producing animals. The residues of these drugs and their metabolites can pose risks for human health. Therefore, a withdrawal time is established to ensure consumer safety so that tissue, milk or eggs from treated animals cannot be harvested for human consumption until enough time has elapsed for the residue levels to decrease to safe concentrations. Part of the process to establish a withdrawal time involves a linear regression to model drug residue depletion over time. This regression model is used to calculate a one-sided, upper tolerance limit for the amount of drug residue remaining in target tissue as a function of time. The withdrawal period is then determined by finding the smallest time so that the upper tolerance limit falls below the maximum residue limit. Observations with measured residue levels at or below the limit of quantitation (LOQ) of the analytical method present a special challenge in the estimation of the tolerance limit. Because values observed below the LOQ are thought to be unreliable, they add in an additional source of uncertainty and, if dealt with improperly or ignored, can introduce bias in the estimation of the withdrawal time. The U.S. Food and Drug Administration (FDA) suggests excluding such data while the European Medicine Agency (EMA) recommends replacing observations below the LOQ with a fixed number, specifically half the value of the LOQ. However, observations below LOQ are technically left censored and these methods are do not effectively address this fact. As an alternative, a regression method accounting for left-censoring is proposed and implemented in order to adequately model residue depletion over time. Furthermore, a method based on generalized (or fiducial) inference is developed to compute a tolerance limit with results from the proposed regression method. A simulation study is then conducted to compare the proposed withdrawal time calculation procedure to the current FDA and EMA approaches. Finally, the proposed procedures are applied to real experimental data.

Neste trabalho, é apresentado vários métodos de diagnóstico para modelos de riscos múltiplos. A vantagem desse modelo é sua flexibilidade em relação aos modelos de risco simples, como, os modelos Weibull e log-logístico, pois acomoda uma grande classe de funções de risco, função de risco não-monótona, por exemplo, forma de "banheira" e curvas multimodal. Alguns métodos de influência, assim como, a influência local, influência local total de um indivíduo são calculadas, analizadas e discutidas. Uma discussão computacional do método do afastamento da verossimilhança, bem como da curvatura normal em influência local são apresentados. Finalmente, um conjunto de dados reais é usado para ilustrar a teoria estudada. Uma análise de resíduo é aplicada para a seleção do modelo apropriado. / In this paperwork is present various diagnostic methods for polyhazard models. Polyhazard models are a flexible family for fitting lifetime data. Their main advantage over the single hazard models, such as the Weibull and the log-logistic models, is to include a large amount of nonmonotone hazard shapes, as bathtub and multimodal curves. Some influence methods, such as the local influence, total local influence of an individual are derived, analyzed and discussed. A discussion of the computation of the likelihood displacement as well as the normal curvature in the local influence method are presented. Finally, an example with real data is given for illustration. A residual analysis is performed in order to select an appropriate model.

Neste trabalho estudou-se a existência de viés de gênero nas escolhas familiares em gastar recursos na educação dos filhos e filhas. Foi obtida uma amostra de 11386 famílias da Pesquisa de Orçamentos Familiares - POF 2002-2003 do Instituto Brasileiro de Geografia e Estatística (IBGE). As famílias tinham filhos e filhas entre 7 e 20 anos de idade, com pelo menos um deles matriculado e sem que outro membro da família, pais ou outros parentes, também estivesse matriculado. Curvas de Engel para as despesas com educação e para gastos com grupos de itens de despesas educacionais foram estimadas. Entre as variáveis explicativas estão o número de filhos e número de filhas matriculados e o número de filhos e o número de filhas não matriculados segundo faixas etárias. Outras variáveis socioeconômicas e demográficas controlaram as demais características familiares. Como 2066 famílias não apresentaram despesas com educação, as curvas de Engel foram estimadas com a utilização de modelos Tobit. As curvas de Engel também foram estimadas por Mínimos Quadrados e os resultados comparados com os obtidos pelos modelos Tobit. A igualdade entre os parâmetros estimados do número de filhos e filhas matriculados em cada faixa etária foi verificada com testes de Wald, isto para os dois procedimentos de estimação utilizados. Os resultados econométricos foram consistentes com a hipótese do trade-off quantidade qualidade existente nas decisões familiares de alocar recursos para a prole, pois o número total de filhos e filhas pressionou as despesas com educação a taxas decrescentes. As análises das despesas com educação, realizadas com os dois procedimentos de estimação, indicaram diferenças significativas no impacto causado por filhos e filhas matriculados nas faixas etárias de 7 a 10 anos e de 15 a 20 anos, com viés pró-feminino. A análise dos gastos com mensalidades escolares com modelos Tobit indicou diferenças significativas para duas faixas etárias, de 11 a 14 anos e de 15 a 20 anos, com viés pró-feminino na última faixa. Com Mínimos Quadrados houve viés pró-feminino na faixa de 15 a 20 anos, nas demais faixas não houve diferenças significativas. Quando foram analisados os gastos com cursos não regulares com um modelo Tobit, foram observadas diferenças significativas de 7 a 10 anos e de 11 a 14 anos, as duas pró-femininas. Ao analisar os gastos com cursos não regulares por Mínimos Quadrados, detectou-se viés pró-masculino na faixa etária de 11 a 14 anos e viés pró-feminino nas demais faixas. Entre as demais estimativas não houve diferenças significativas ou elas foram pró-femininas, independente do procedimento de estimação. Os resultados indicam que a formação das jovens não sofreu discriminação no que tange à disposição das famílias de gastarem com a formação de sua prole, salvo as duas exceções mencionadas. Estes resultados mostraram-se consistentes com o cenário favorável à escolarização feminina no Brasil já apontado em outros estudos. / This study investigated the gender bias presence within the family choices about spending resourses with their children\'s education. A sample of 11386 families was obtained from Household Budget Survey - POF 2002-2003, of the Brazilian Institute of Geography and Statistic - IBGE. The families had children between 7 and 20 years old, with at least one of them enrolled, considering that no other member of the family, parents or relatives, was enrolled. Engel curves for the overall expenses and for expenditures with educational item groups were estimated. The number of sons and daughters enrolled and the number of sons and daughters not enrolled according to age level are among the explanatory variables. Other socioeconomics and demographic variables controlled the rest of the family characteristics. As 2066 families did not showed expenses with education, the Engel curves were estimated using Tobit models. The Engel curves were also estimated by the method of Least Squares and the results were compared to the ones obtained by Tobit model. The equity between the estimated parameters of the number of sons and daughters enrolled in each age level was verified with Wald test for both the estimation procedures used. The econometric results were consistent with the hypothesis of quantity-quality trade-off which exists within the family decisions when allocating resourses for their children because the total number of children pressed the educational expenditures at decreasing rates. The analysis of total expenses with education, accomplished by both the estimation procedures, indicated significant differences in the impact caused by sons and daughters enrolled at the age level from 7 to 10 and from 15 to 20 with pro-female bias. The analysis of the expenditures with school monthly fees with Tobit models indicated significant differences for both age levels, from 11 to 14 and from 15 to 20, with a pro-female bias in the latter level. With the Minimum Square method there was a pro-female bias in the level of age from 15 to 20 years old, in the other levels there were no significant differences. When the expenditures with non regular courses were analyzed with Tobit model, significant differences were observed from 7 to 10 and from 11 to 14 years old, both pro-female. When the expenditures with non regular courses were analyzed with the Minimum Square method a pro-male bias was detected in the age level from 11 to 14 and a pro-female bias in the other levels. Among the other estimations there were no significant differences or they were pro-female, despite the estimation procedure. The results show that the young girls\' education did not suffer any discrimination related to the family disposition of spending with their children\'s education, apart from the two exceptions mentioned. These results are consistent with the favorable scenario to the female education in Brazil already pointed out in other studies.

Neste trabalho considera-se uma reparametrização no modelo log-gama generalizado para a inclusão de dados com sobreviventes de longa duração. Os modelos tentam estimar separadamente os efeitos das covariáveis na aceleração ou desaceleração no tempo e na fração de sobreviventes que é a proporção da população para o qual o evento não ocorre. A função logística é usada para o modelo de regressão com fração de cura. Os parâmetros do modelo, serão estimados através do método de máxima verossimilhança. Alguns métodos de influência, como a influência local e a influência local total de um indivíduo, serão introduzidos, calculados, analisados e discutidos. Finalmente, um conjunto de dados médicos será analisado sob o modelo log-gama generalizado com fração de cura. Uma análise de resíduos será executada para verificar a qualidade de ajuste do modelo. / In this work the generalized log-gama model is modified for possibility that long-term survivors are present in the data . The models attempt to estimate separately the effects of covariates on the accelaration/decelaration of the timing of a given event and surviving fraction; that is, the proportion of the population for which the event never occurs. The logistic function is used for the regression model of the surviving fraction. Inference for the model parameters is considered via maximum likelihood. Some influence methods, such as the local influence, total local influence of an individual are derived, analyzed and discussed. Finally, a data set from the medical area is analyzed under log-gama generalized mixture model. A residual analysis is performed in order to select an appropriate model.

In this paper we address the problem of negative estimates of willingness to pay. We find that there exist a number of goods and services, especially in the fields of marketing and environmental valuation, for which only zero or positive WTP is meaningful. For the valuation of these goods an econometric model for the analysis of repeated dichotomous choice data is proposed. Our model restricts the domain of the estimates of WTP to strictly positive values, while also allowing for the detection of zero WTP. The model is tested on a simulated and a real data set.

In the real world, many problems are continuous in nature. In some cases, finding the global solutions for these problems is di±cult. The reason is that the problem's objective function is non convex, nor concave and even not differentiable. Tackling these problems is often computationally too expensive. Although the development in computer technologies are increasing the speed of computations, this often is not adequate, particularly if the size of the problem's instance are large. Applying exact methods on some problems may necessitate their linearisation. Several new ideas using heuristic approaches have been considered particularly since they tackle the problems within reasonable computational time and give an approximate solution. In this thesis, the variable neighbourhood search (VNS) metaheuristic (the framework for building heuristic) has been considered. Two variants of variable neighbourhood search metaheuristic have been developed, continuous variable neighbourhood search and reformulation descent variable neighbourhood search. The GLOB-VNS software (Drazic et al., 2006) hybridises the Microsoft Visual Studio C++ solver with variable neighbourhood search metaheuristics. It has been used as a starting point for this research and then adapted and modified for problems studied in this thesis. In fact, two problems have been considered, censored quantile regression and the circle packing problem. The results of this approach for censored quantile regression outperforms other methods described in the literature, and the near-optimal solutions are obtained in short running computational time. In addition, the reformulation descent variable neighbourhood search variant in solving circle packing problems is developed and the computational results are provided.