In this dissertation, we approach three different contributions in measurement error model (MEM). Initially, we carry out maximum penalized likelihood inference in MEMs under the normality assumption. The methodology is based on the method proposed by Firth (1993), which can be used to improve some asymptotic properties of the maximum likelihood estimators. In the second contribution, we develop two new estimation methods based on generalized fiducial inference for the precision parameters and the variability product under the Grubbs model considering the two-instrument case. One method is based on a fiducial generalized pivotal quantity and the other one is built on the method of the generalized fiducial distribution. Comparisons with two existing approaches are reported. Finally, we propose to study inference in a heteroscedastic MEM with known error variances. Instead of the normal distribution for the random components, we develop a model that assumes a skew-t distribution for the true covariate and a centered Students t distribution for the error terms. The proposed model enables to accommodate skewness and heavy-tailedness in the data, while the degrees of freedom of the distributions can be different. We use the maximum likelihood method to estimate the model parameters and compute them via an EM-type algorithm. All proposed methodologies are assessed numerically through simulation studies and illustrated with real datasets extracted from the literature. / Neste trabalho abordamos três contribuições diferentes em modelos com erros de medição (MEM). Inicialmente estudamos inferência pelo método de máxima verossimilhança penalizada em MEM sob a suposição de normalidade. A metodologia baseia-se no método proposto por Firth (1993), o qual pode ser usado para melhorar algumas propriedades assintóticas de os estimadores de máxima verossimilhança. Em seguida, propomos construir dois novos métodos de estimação baseados na inferência fiducial generalizada para os parâmetros de precisão e a variabilidade produto no modelo de Grubbs para o caso de dois instrumentos. O primeiro método é baseado em uma quantidade pivotal generalizada fiducial e o outro é baseado no método da distribuição fiducial generalizada. Comparações com duas abordagens existentes são reportadas. Finalmente, propomos estudar inferência em um MEM heterocedástico em que as variâncias dos erros são consideradas conhecidas. Nós desenvolvemos um modelo que assume uma distribuição t-assimétrica para a covariável verdadeira e uma distribuição t de Student centrada para os termos dos erros. O modelo proposto permite acomodar assimetria e cauda pesada nos dados, enquanto os graus de liberdade das distribuições podem ser diferentes. Usamos o método de máxima verossimilhança para estimar os parâmetros do modelo e calculá-los através de um algoritmo tipo EM. Todas as metodologias propostas são avaliadas numericamente em estudos de simulação e são ilustradas com conjuntos de dados reais extraídos da literatura

<p>    Adaptive MCMC is advantageous over traditional MCMC due to its ability to automatically adjust its proposal distributions during the sampling process, providing improved sampling efficiency and faster convergence to the target distribution, especially in complex or high-dimensional problems. However, designing and validating the adaptive scheme cautiously is crucial to ensure algorithm validity and prevent the introduction of biases. This dissertation focuses on the use of Adaptive MCMC for deep learning, specifically addressing the mode collapse issue in Generative Adversarial Networks (GANs) and implementing Fiducial inference, and its application to Causal inference in individual treatment effect problems.</p> <p><br></p> <p>    First, GAN was recently introduced in the literature as a novel machine learning method for training generative models. However, GAN is very difficult to train due to the issue of mode collapse, i.e., lack of diversity among generated data. We figure out the reason why GAN suffers from this issue and lay out a new theoretical framework for GAN based on randomized decision rules such that the mode collapse issue can be overcome essentially. Under the new theoretical framework, the discriminator converges to a fixed point while the generator converges to a distribution at the Nash equilibrium.</p> <p><br></p> <p>    Second, Fiducial inference was generally considered as R.A. Fisher's a big blunder, but the goal he initially set, <em>making inference for the uncertainty of model parameters on the basis of observations</em>, has been continually pursued by many statisticians. By leveraging on advanced statistical computing techniques such as stochastic approximation Markov chain Monte Carlo, we develop a new statistical inference method, the so-called extended Fiducial inference, which achieves the initial goal of fiducial inference. </p> <p><br></p> <p>    Lastly, estimating ITE is important for decision making in various fields, particularly in health research where precision medicine is being investigated. Conditional average treatment effect (CATE) is often used for such purpose, but uncertainty quantification and explaining the variability of predicted ITE is still needed for fair decision making. We discuss using extended Fiducial inference to construct prediction intervals for ITE, and introduces a double neural net algorithm for efficient prediction and estimation of nonlinear ITE.</p>