O objetivo deste trabalho é apresentar a eficiência dos estimadores quando são usados os algoritmos SIMEX e EM nos modelos de regressão lineares mistos com erros nas variáveis, numa primeira etapa apresentamos o análise do algoritmo SIMEX num modelo de regressão simples com a finalidade de ver seus vantagens, numa segunda etapa apresentamos o modelos de regressão linear misto sem erros nas variáveis com a finalidade de observar seus estimadores, numa terceira etapa consideramos os algoritmos SIMEX e EM num modelo de regressão linear misto com erros nas variáveis, observando os estimadores obtidos e comparando-los com aqueles obtidos no modelo de regressão linear misto sem erros nas variáveis, com a finalidade de ver se os estimadores obtidos com os dois algoritmos são razoáveis, fazendo também uma comparação entre os estimadores obtidos por ambos algoritmos. Os programas foram feitos no pacote OX para a obtenção das estimativas dos algoritmos propostos.

O objetivo deste trabalho é apresentar a eficiência dos estimadores quando são usados os algoritmos SIMEX e EM nos modelos de regressão lineares mistos com erros nas variáveis, numa primeira etapa apresentamos o análise do algoritmo SIMEX num modelo de regressão simples com a finalidade de ver seus vantagens, numa segunda etapa apresentamos o modelos de regressão linear misto sem erros nas variáveis com a finalidade de observar seus estimadores, numa terceira etapa consideramos os algoritmos SIMEX e EM num modelo de regressão linear misto com erros nas variáveis, observando os estimadores obtidos e comparando-los com aqueles obtidos no modelo de regressão linear misto sem erros nas variáveis, com a finalidade de ver se os estimadores obtidos com os dois algoritmos são razoáveis, fazendo também uma comparação entre os estimadores obtidos por ambos algoritmos. Os programas foram feitos no pacote OX para a obtenção das estimativas dos algoritmos propostos.

Small area estimation is an arena that has seen rapid development in the past 50 years, due to its widespread applicability in government projects, marketing research and many other areas. However, it is often difficult to obtain error-free data for this purpose. In this dissertation, each project describes a model used for small area estimation in which the covariates are measured with error. We applied different methods of bias correction to improve the estimates of the parameter of interest in the small areas. There is a variety of methods available for bias correction of estimates in the presence of measurement error. We applied the simulation extrapolation (SIMEX), ordinary corrected scores and Monte Carlo corrected scores methods of bias correction in the Fay-Herriot model, and investigated the performance of the bias-corrected estimators. The performance of the estimators in the presence of non-normal measurement error and of the SIMEX estimator in the presence of non-additive measurement error was also studied. For each of these situations, we presented simulation studies to observe the performance of the proposed correction procedures. In addition, we applied our proposed methodology to analyze a real life, nontrivial data set and present the results. We showed that the Lohr-Ybarra estimator is slightly inefficient and that applying methods of bias correction like SIMEX, corrected scores or Monte Carlo corrected scores (MCCS) increases the efficiency of the small area estimates. In particular, we showed that the simulation based bias correction methods like SIMEX and MCCS provide a greater gain in efficiency. We also showed that the SIMEX method of bias correction is robust with respect to departures from normality or additivity of measurement error. We showed that the MCCS method is robust with respect to departure from normality of measurement error.

Modelos com erros de medição têm recebido a atenção de vários pesquisadores das mais diversas áreas de conhecimento. O principal objetivo desta dissertação consiste no estudo de um modelo funcional com erros de medição heteroscedásticos na presença de réplicas das observações. O modelo proposto estende resultados encontrados na literatura na medida em que as réplicas são parte do modelo, ao contrário de serem utilizadas para estimação das variâncias, doravante tratadas como conhecidas. Alguns procedimentos de estimação tais como o método de máxima verossimilhança, o método dos momentos e o método de extrapolação da simulação (SIMEX) na versão empírica são apresentados. Além disso, propõe-se o teste da razão de verossimilhanças e o teste de Wald com o objetivo de testar algumas hipóteses de interesse relacionadas aos parâmetros do modelo adotado. O comportamento dos estimadores de alguns parâmetros e das estatísticas propostas (resultados assintóticos) são analisados por meio de um estudo de simulação de Monte Carlo, utilizando-se diferentes números de réplicas. Por fim, a proposta é exemplificada com um conjunto de dados reais. Toda parte computacional foi desenvolvida em linguagem R (R Development Core Team, 2011) / Measurement error models have received the attention of many researchers of several areas of knowledge. The aim of this dissertation is to study a functional heteroscedastic measurement errors model with replicated observations. The proposed model extends results from the literature in that replicas are part of the model, as opposed to being used for estimation of the variances, now treated as known. Some estimation procedures such as maximum likelihood method, the method of moments and the empirical simulation-extrapolation method (SIMEX) are presented. Moreover, it is proposed the likelihood ratio test and Wald test in order to test hypotheses of interest related to the model parameters used. The behavior of the estimators of some parameters and statistics proposed (asymptotic results) are analyzed through Monte Carlo simulation study using different numbers of replicas. Finally, the proposal is illustrated with a real data set. The computational part was developed in R language (R Development Core Team, 2011)

Modelos com erros de medição têm recebido a atenção de vários pesquisadores das mais diversas áreas de conhecimento. O principal objetivo desta dissertação consiste no estudo de um modelo funcional com erros de medição heteroscedásticos na presença de réplicas das observações. O modelo proposto estende resultados encontrados na literatura na medida em que as réplicas são parte do modelo, ao contrário de serem utilizadas para estimação das variâncias, doravante tratadas como conhecidas. Alguns procedimentos de estimação tais como o método de máxima verossimilhança, o método dos momentos e o método de extrapolação da simulação (SIMEX) na versão empírica são apresentados. Além disso, propõe-se o teste da razão de verossimilhanças e o teste de Wald com o objetivo de testar algumas hipóteses de interesse relacionadas aos parâmetros do modelo adotado. O comportamento dos estimadores de alguns parâmetros e das estatísticas propostas (resultados assintóticos) são analisados por meio de um estudo de simulação de Monte Carlo, utilizando-se diferentes números de réplicas. Por fim, a proposta é exemplificada com um conjunto de dados reais. Toda parte computacional foi desenvolvida em linguagem R (R Development Core Team, 2011) / Measurement error models have received the attention of many researchers of several areas of knowledge. The aim of this dissertation is to study a functional heteroscedastic measurement errors model with replicated observations. The proposed model extends results from the literature in that replicas are part of the model, as opposed to being used for estimation of the variances, now treated as known. Some estimation procedures such as maximum likelihood method, the method of moments and the empirical simulation-extrapolation method (SIMEX) are presented. Moreover, it is proposed the likelihood ratio test and Wald test in order to test hypotheses of interest related to the model parameters used. The behavior of the estimators of some parameters and statistics proposed (asymptotic results) are analyzed through Monte Carlo simulation study using different numbers of replicas. Finally, the proposal is illustrated with a real data set. The computational part was developed in R language (R Development Core Team, 2011)

Hydrologic and climate models predict variables through a simplification of the underlying complex natural processes. Model development involves minimising predictive uncertainty. Predictive uncertainty arises from three broad sources which are measurement error in observed responses, uncertainty of input variables and model structural error. This thesis introduces ways to improve predictive accuracy of hydroclimatic models by considering input and structural uncertainties. The specific methods developed to reduce the uncertainty because of erroneous inputs and model structural errors are outlined below. The uncertainty in hydrological model inputs, if ignored, introduces systematic biases in the parameters estimated. This thesis presents a method, known as simulation extrapolation (SIMEX), to ascertain the extent of parameter bias. SIMEX starts by generating a series of alternate inputs by artificially adding white noise in increasing multiples of the known input error variance. The resulting alternate parameter sets allow formulation of an empirical relationship between their values and the level of noise present. SIMEX is based on the theory that the trend in alternate parameters can be extrapolated back to the notional error free zone. The case study relates to erroneous sea surface temperature anomaly (SSTA) records used as input variables of a linear model to predict the Southern Oscillation Index (SOI). SIMEX achieves a reduction in residual errors from the SOI prediction. Besides, a hydrologic application of SIMEX is demonstrated by a synthetic simulation within a three-parameter conceptual rainfall runoff model. This thesis next advocates reductions of structural uncertainty of any single model by combining multiple alternative model responses. Current approaches for combining hydroclimatic forecasts are generally limited to using combination weights that remain static over time. This research develops a methodology for combining forecasts from multiple models in a dynamic setting as an improvement of over static weight combination. The model responses are mixed on a pair wise basis using mixing weights that vary in time reflecting the persistence of individual model skills. The concept is referred here as the pair wise dynamic weight combination. Two approaches for forecasting the dynamic weights are developed. The first of the two approaches uses a mixture of two basis distributions which are three category ordered logistic regression model and a generalised linear autoregressive model. The second approach uses a modified nearest neighbour approach to forecast the future weights. These alternatives are used to first combine a univariate response forecast, the NINO3.4 SSTA index. This is followed by a similar combination, but for the entire global gridded SSTA forecast field. Results from these applications show significant improvements being achieved due to the dynamic model combination approach. The last application demonstrating the dynamic combination logic, uses the dynamically combined multivariate SSTA forecast field as the basis of developing multi-site flow forecasts in the Namoi River catchment in eastern Australia. To further reduce structural uncertainty in the flow forecasts, three forecast models are formulated and the dynamic combination approach applied again. The study demonstrates that improved SSTA forecast (due to dynamic combination) in turn improves all three flow forecasts, while the dynamic combination of the three flow forecasts results in further improvements.

Hydrologic and climate models predict variables through a simplification of the underlying complex natural processes. Model development involves minimising predictive uncertainty. Predictive uncertainty arises from three broad sources which are measurement error in observed responses, uncertainty of input variables and model structural error. This thesis introduces ways to improve predictive accuracy of hydroclimatic models by considering input and structural uncertainties. The specific methods developed to reduce the uncertainty because of erroneous inputs and model structural errors are outlined below. The uncertainty in hydrological model inputs, if ignored, introduces systematic biases in the parameters estimated. This thesis presents a method, known as simulation extrapolation (SIMEX), to ascertain the extent of parameter bias. SIMEX starts by generating a series of alternate inputs by artificially adding white noise in increasing multiples of the known input error variance. The resulting alternate parameter sets allow formulation of an empirical relationship between their values and the level of noise present. SIMEX is based on the theory that the trend in alternate parameters can be extrapolated back to the notional error free zone. The case study relates to erroneous sea surface temperature anomaly (SSTA) records used as input variables of a linear model to predict the Southern Oscillation Index (SOI). SIMEX achieves a reduction in residual errors from the SOI prediction. Besides, a hydrologic application of SIMEX is demonstrated by a synthetic simulation within a three-parameter conceptual rainfall runoff model. This thesis next advocates reductions of structural uncertainty of any single model by combining multiple alternative model responses. Current approaches for combining hydroclimatic forecasts are generally limited to using combination weights that remain static over time. This research develops a methodology for combining forecasts from multiple models in a dynamic setting as an improvement of over static weight combination. The model responses are mixed on a pair wise basis using mixing weights that vary in time reflecting the persistence of individual model skills. The concept is referred here as the pair wise dynamic weight combination. Two approaches for forecasting the dynamic weights are developed. The first of the two approaches uses a mixture of two basis distributions which are three category ordered logistic regression model and a generalised linear autoregressive model. The second approach uses a modified nearest neighbour approach to forecast the future weights. These alternatives are used to first combine a univariate response forecast, the NINO3.4 SSTA index. This is followed by a similar combination, but for the entire global gridded SSTA forecast field. Results from these applications show significant improvements being achieved due to the dynamic model combination approach. The last application demonstrating the dynamic combination logic, uses the dynamically combined multivariate SSTA forecast field as the basis of developing multi-site flow forecasts in the Namoi River catchment in eastern Australia. To further reduce structural uncertainty in the flow forecasts, three forecast models are formulated and the dynamic combination approach applied again. The study demonstrates that improved SSTA forecast (due to dynamic combination) in turn improves all three flow forecasts, while the dynamic combination of the three flow forecasts results in further improvements.

In ecological population management, years of animal counts are fit to nonlinear, dynamic models (e.g. the Ricker model) because the values of the parameters are of interest. The yearly counts are subject to measurement error, which inevitably leads to biased estimates and adversely affects inference if ignored. In the literature, often convenient distribution assumptions are imposed, readily available estimated measurement error variances are not utilized, or the measurement error is ignored entirely. In this thesis, ways to estimate the parameters of the Ricker model and perform inference while accounting for measurement error are investigated where distribution assumptions are minimized and estimated measurement error variances are utilized. To these ends, SIMEX and modified estimating equations (MEE) rather than likelihood methods are investigated for data on the abundance and log-abundance scales, and how inference is done via the parametric bootstrap and estimated standard errors from the modified estimating equations is shown. Subsequently, simulation studies are performed on the log-abundance scale under varying parameter values to learn how levels of measurement error variances (ranging from the realistically low value of 0.0025 to unrealistically high value of 0.025 ) affects the estimators and inference when measurement error is ignored, and how the methods perform accounting for it. It was found that the bias induced by measurement error depends on the true value of the parameter. Furthermore, the performances of SIMEX and MEE are associated with the true value of a and the level of measurement error variance. In particular, both methods perform best for a > 1 and low to moderate levels of measurement error variance, with the MEE estimators having high standard error and often poorer performance than those from SIMEX. It was also found that the MEE estimators contain singularities which attribute to its low precision and erratic behavior. These methods were then applied to actual moose count data with sample size more than double that of the simulations. It was found that both the SIMEX and MEE estimators performed well suggesting that sample size contributes to previous poor behavior.

Neste trabalho estudamos o modelo de regressão logística com erro de medida nas covariáveis. Abordamos as metodologias de estimação de máxima pseudoverossimilhança pelo algoritmo EM-Monte Carlo, calibração da regressão, SIMEX e naïve (ingênuo), método este que ignora o erro de medida. Comparamos os métodos em relação à estimação, através do viés e da raiz do erro quadrático médio, e em relação à predição de novas observações, através das medidas de desempenho sensibilidade, especificidade, verdadeiro preditivo positivo, verdadeiro preditivo negativo, acurácia e estatística de Kolmogorov-Smirnov. Os estudos de simulação evidenciam o melhor desempenho do método de máxima pseudoverossimilhança na estimação. Para as medidas de desempenho na predição não há diferença entre os métodos de estimação. Por fim, utilizamos nossos resultados em dois conjuntos de dados reais de diferentes áreas: área médica, cujo objetivo está na estimação da razão de chances, e área financeira, cujo intuito é a predição de novas observações. / We study the logistic model when explanatory variables are measured with error. Three estimation methods are presented, namely maximum pseudo-likelihood obtained through a Monte Carlo expectation-maximization type algorithm, regression calibration, SIMEX and naïve, which ignores the measurement error. These methods are compared through simulation. From the estimation point of view, we compare the different methods by evaluating their biases and root mean square errors. The predictive quality of the methods is evaluated based on sensitivity, specificity, positive and negative predictive values, accuracy and the Kolmogorov-Smirnov statistic. The simulation studies show that the best performing method is the maximum pseudo-likelihood method when the objective is to estimate the parameters. There is no difference among the estimation methods for predictive purposes. The results are illustrated in two real data sets from different application areas: medical area, whose goal is the estimation of the odds ratio, and financial area, whose goal is the prediction of new observations.

Neste trabalho estudamos o modelo de regressão logística com erro de medida nas covariáveis. Abordamos as metodologias de estimação de máxima pseudoverossimilhança pelo algoritmo EM-Monte Carlo, calibração da regressão, SIMEX e naïve (ingênuo), método este que ignora o erro de medida. Comparamos os métodos em relação à estimação, através do viés e da raiz do erro quadrático médio, e em relação à predição de novas observações, através das medidas de desempenho sensibilidade, especificidade, verdadeiro preditivo positivo, verdadeiro preditivo negativo, acurácia e estatística de Kolmogorov-Smirnov. Os estudos de simulação evidenciam o melhor desempenho do método de máxima pseudoverossimilhança na estimação. Para as medidas de desempenho na predição não há diferença entre os métodos de estimação. Por fim, utilizamos nossos resultados em dois conjuntos de dados reais de diferentes áreas: área médica, cujo objetivo está na estimação da razão de chances, e área financeira, cujo intuito é a predição de novas observações. / We study the logistic model when explanatory variables are measured with error. Three estimation methods are presented, namely maximum pseudo-likelihood obtained through a Monte Carlo expectation-maximization type algorithm, regression calibration, SIMEX and naïve, which ignores the measurement error. These methods are compared through simulation. From the estimation point of view, we compare the different methods by evaluating their biases and root mean square errors. The predictive quality of the methods is evaluated based on sensitivity, specificity, positive and negative predictive values, accuracy and the Kolmogorov-Smirnov statistic. The simulation studies show that the best performing method is the maximum pseudo-likelihood method when the objective is to estimate the parameters. There is no difference among the estimation methods for predictive purposes. The results are illustrated in two real data sets from different application areas: medical area, whose goal is the estimation of the odds ratio, and financial area, whose goal is the prediction of new observations.