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ABSTRACT ROBUST CONTROL CHARTS &Ccedil / etiny&uuml / rek, Aysun M. Sc., Department of Statistics Supervisor: Dr. BariS S&uuml / r&uuml / c&uuml / Co-Supervisor: Assoc. Prof. Dr. Birdal Senoglu December 2006, 82 pages Control charts are one of the most commonly used tools in statistical process control. A prominent feature of the statistical process control is the Shewhart control chart that depends on the assumption of normality. However, violations of underlying normality assumption are common in practice. For this reason, control charts for symmetric distributions for both long- and short-tailed distributions are constructed by using least squares estimators and the robust estimators -modified maximum likelihood, trim, MAD and wave. In order to evaluate the performance of the charts under the assumed distribution and investigate robustness properties, the probability of plotting outside the control limits is calculated via Monte Carlo simulation technique.

The main objective of this dissertation is the development of CUSUM procedures based on signed and unsigned sequential ranks. These CUSUMs can be applied to detect changes in the location or dispersion of a process. The signed and unsigned sequential rank CUSUMs are distribution-free and robust against the effect of outliers in the data. The only assumption that these CUSUMs require is that the in-control distribution is symmetric around a known location parameter. These procedures specifically do not require the existence of any higher order moments. Another advantage of these CUSUMs is that Monte Carlo simulation can readily be applied to deliver valid estimates of control limits, irrespective of what the underlying distribution may be. Other objectives of this dissertation include a brief discussion of the results and refinements of the CUSUM in the literature. We justify the use of a signed sequential rank statistic. Also, we evaluate the relative efficiency of the suggested procedure numerically and provide three real-world applications from the engineering and financial industries. / MSc (Risk Analysis), North-West University, Potchefstroom Campus, 2015

The main objective of this dissertation is the development of CUSUM procedures based on signed and unsigned sequential ranks. These CUSUMs can be applied to detect changes in the location or dispersion of a process. The signed and unsigned sequential rank CUSUMs are distribution-free and robust against the effect of outliers in the data. The only assumption that these CUSUMs require is that the in-control distribution is symmetric around a known location parameter. These procedures specifically do not require the existence of any higher order moments. Another advantage of these CUSUMs is that Monte Carlo simulation can readily be applied to deliver valid estimates of control limits, irrespective of what the underlying distribution may be. Other objectives of this dissertation include a brief discussion of the results and refinements of the CUSUM in the literature. We justify the use of a signed sequential rank statistic. Also, we evaluate the relative efficiency of the suggested procedure numerically and provide three real-world applications from the engineering and financial industries. / MSc (Risk Analysis), North-West University, Potchefstroom Campus, 2015

Cordeiro e Andrade (2009) incorporam a ideia de variável resposta transformada ao modelo GARMA, autorregressivo e de médias móveis generalizado, introduzido por Benjamin et al. (2003), desenvolvendo assim, o modelo TGARMA, autorregressivo e de médias móveis generalizado transformado. O objetivo do presente trabalho é desenvolver o modelo TGARMA introduzido por Cordeiro e Andrade (2009) para distribuições condicionais simétricas contínuas com e sem heteroscedasticidade e uma possível função não linear para os parâmetros de regressão. Ao longo desta tese derivamos um processo iterativo para estimar os parâmetros desses modelos por máxima verossimilhança. Nós produzimos uma fórmula simples para estimar o parâmetro que define a transformação da variável resposta para uma subclasse de modelos. Fornecemos os momentos para a variável dependente original. Para o modelo homoscedástico, discutimos inferência de alguns parâmetros, propomos uma análise de diagnóstico e a definição de um resíduo padronizado. Finalmente, para ilustrar a teoria desenvolvida, tanto no caso homoscedástico quanto no caso heteroscedástico, utilizamos conjuntos de dados reais e avaliamos os resultados desenvolvidos por meio de estudos de simulação. / Cordeiro and Andrade (2009) incorporate the idea of transforming the response variable to the GARMA model, generalized autoregressive moving average, introduced by Benjamin et al. (2003), thus developing the TGARMA model, transformed generalized autoregressive moving average. The goal of this thesis is to develop the TGARMA model introduced by Cordeiro and Andrade (2009) for symmetric continuous conditional distributions and a possible non-linear structure for the mean that enables the fitting of a wide range of models to several data types. When the assumption of homoscedasticity is not verified, heteroscedastic models are proposed. Throughout this thesis, we derive an iterative process for fitting the parameters of the models by maximum likelihood. We produce a simple formula to estimate the parameter which defines the transformation of the response variable and the moments of the original dependent variable which generalize previous published results. For the homoscedastic model, we discuss inference, we propose a diagnostic analysis and define a standardized residuals. Finally, to illustrate the theory developed, we use real data sets and we evaluate the results developed through simulations studies.

Cordeiro e Andrade (2009) incorporam a ideia de variável resposta transformada ao modelo GARMA, autorregressivo e de médias móveis generalizado, introduzido por Benjamin et al. (2003), desenvolvendo assim, o modelo TGARMA, autorregressivo e de médias móveis generalizado transformado. O objetivo do presente trabalho é desenvolver o modelo TGARMA introduzido por Cordeiro e Andrade (2009) para distribuições condicionais simétricas contínuas com e sem heteroscedasticidade e uma possível função não linear para os parâmetros de regressão. Ao longo desta tese derivamos um processo iterativo para estimar os parâmetros desses modelos por máxima verossimilhança. Nós produzimos uma fórmula simples para estimar o parâmetro que define a transformação da variável resposta para uma subclasse de modelos. Fornecemos os momentos para a variável dependente original. Para o modelo homoscedástico, discutimos inferência de alguns parâmetros, propomos uma análise de diagnóstico e a definição de um resíduo padronizado. Finalmente, para ilustrar a teoria desenvolvida, tanto no caso homoscedástico quanto no caso heteroscedástico, utilizamos conjuntos de dados reais e avaliamos os resultados desenvolvidos por meio de estudos de simulação. / Cordeiro and Andrade (2009) incorporate the idea of transforming the response variable to the GARMA model, generalized autoregressive moving average, introduced by Benjamin et al. (2003), thus developing the TGARMA model, transformed generalized autoregressive moving average. The goal of this thesis is to develop the TGARMA model introduced by Cordeiro and Andrade (2009) for symmetric continuous conditional distributions and a possible non-linear structure for the mean that enables the fitting of a wide range of models to several data types. When the assumption of homoscedasticity is not verified, heteroscedastic models are proposed. Throughout this thesis, we derive an iterative process for fitting the parameters of the models by maximum likelihood. We produce a simple formula to estimate the parameter which defines the transformation of the response variable and the moments of the original dependent variable which generalize previous published results. For the homoscedastic model, we discuss inference, we propose a diagnostic analysis and define a standardized residuals. Finally, to illustrate the theory developed, we use real data sets and we evaluate the results developed through simulations studies.

In this thesis, pairwise multiple comparisons and multiple comparisons with a control are studied when the observations have short-tailed symmetric distributions. Under non-normality, the testing procedure is given and Huber estimators, trimmed mean with winsorized standard deviation, modified maximum likelihood estimators and ordinary sample mean and sample variance used in this procedure are reviewed. Finally, robustness properties of the stated estimators are compared with each other and it is shown that the test based on the modified maximum likelihood estimators has better robustness properties under short-tailed symmetric distribution.

The study of properties of probability distributions has always been a persistent theme of statistics and of applied probability. This thesis deals with an investigation of distribution functions under the following two topics: (i) characterization of distributions based on record values and order statistics, (ii) properties of the skew-t distribution. Within the extensive characterization literature there are several results involving properties of record values and order statistics. Although there have been many well known results already developed, it is still of great interest to find new characterization of distributions based on record values and order statistics. In the first part, we provide the conditional distribution of any record value given the maximum order statistics and study characterizations of distributions based on record values and the maximum order statistics. We also give some characterizations of the mean value function within the class of order statistics point processes, by using certain relations between the conditional moments of the jump times or current lives. These results can be applied to characterize the uniform distribution using the sequence of order statistics, and the exponential distribution using the sequence of record values, respectively. Azzalini (1985, 1986) introduced the skew-normal distribution which includes the normal distribution and has some properties like the normal and yet is skew. This class of distributions is useful in studying robustness and for modeling skewness. Since then, skew-symmetric distributions have been proposed by many authors. In the second part, the so-called generalized skew-t distribution is defined and studied. Examples of distributions in this class, generated by the ratio of two independent skew-symmetric distributions, are given. We also investigate properties of the skew-symmetric distribution.

Submitted by (ana.araujo@ufrpe.br) on 2016-05-25T15:49:21Z No. of bitstreams: 1 Carlos Sergio Araujo dos Santos.pdf: 1655656 bytes, checksum: 7687ae013a71aadd4e9e9bf4783a172f (MD5) / Made available in DSpace on 2016-05-25T15:49:21Z (GMT). No. of bitstreams: 1 Carlos Sergio Araujo dos Santos.pdf: 1655656 bytes, checksum: 7687ae013a71aadd4e9e9bf4783a172f (MD5) Previous issue date: 2010-01-15 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / Box and Cox (1964) developed a numerical procedure to transform the response variable such that the transformed variable should be as closed as possible to the normal distribution. The introduction of a new class of non linear symetric transformed models aims to extended the Box and Cox models to a general class of symetric models. The new class of models inclued all the continuos symmetric distributions with a possible non linear structure to the mean, making possible the use of the new class of regression models. It was applied in the estimate of volumes of the Eucalyptus tereticornis clones, with 7,5 years, planted in the Experimental Station of Araripe of the Agronomic Institute of Pernambuco (IPA), in the municipality of Araripina, in the semiarid of Pernambuco. The non linear model used as pattern was the Schumacher and Hall model. The results indicates that the transformed model with t-Student erros with two degrees of freedon adjusted better to the data set. / Box e Cox (1964) desenvolveram um procedimento numérico para escolher uma transformação da resposta tal que a distribuição da variável transformada esteja o mais próximo possível da distribuição normal. A introdução de uma nova classe de modelos simétricos transformados não lineares visa estender os modelos de Box e Cox para uma classe geral dos modelos simétricos. Esta nova classe de modelos inclui todas as distribuições contínuas simétricas com uma possível estrutura não linear para a média e capacitando o ajustamento de uma larga extensão de modelos para vários tipos de dados. Para ilustrar a utilidade dessa nova classe de modelos de regressão foi realizada uma aplicação na estimativa dos volumes de clones de Eucalyptus tereticornis com 7,5 anos oriundos de um experimento que está sendo realizado no Campo Experimental do Araripe do Instituto Agronômico de Pernambuco (IPA), localizado no Município de Araripina, no semiárido Pernambucano. O modelo não-linear utilizado para explicar os dados foi o modelo Schumacher-Hall. Diante dos resultados obtidos se concluí que o modelo transformado com erros t-Student com dois graus de liberdade foi o que melhor se ajustou os dados.

Submitted by (ana.araujo@ufrpe.br) on 2016-07-07T14:09:13Z No. of bitstreams: 1 Katia Pires do Nascimento.pdf: 303593 bytes, checksum: 6b936f81d2b21d770e2224c3fbdd07c1 (MD5) / Made available in DSpace on 2016-07-07T14:09:13Z (GMT). No. of bitstreams: 1 Katia Pires do Nascimento.pdf: 303593 bytes, checksum: 6b936f81d2b21d770e2224c3fbdd07c1 (MD5) Previous issue date: 2010-02-25 / This manuscript has two aims. First, we derive general matrix formulae to Bartlett–type correction to the score statistic in a class of heteroscedastic symmetric nonlinear regression models, with link functions any for both mean and dispersion parameter. In the second part Monte Carlo simulations are also performed to assess the influence of the correction in the models studied. / Essa dissertação tem dois objetivos. O primeiro é a obtenção de expressões matriciais para o fator de correção tipo–Bartlett para a estatística escore nos modelos não–lineares simétricos heteroscedásticos, com funções de ligação quaisquer para a média e para o parâmetro de dispersão. O segundo é apresentar resultados de simulação de forma a verificar a influência da correção nos modelos em estudo.