Maximum likelihood estimation of measurement error models based on the Monte Carlo EM algorithm
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Likelihood based estimation of stochastic models when one of the explanatory variables is masked by measurement error, is presented. Special methods are required to estimate the parameters of a model with one or more explanatory variables that are measured with error. In such models, the variable measured with error is unobservable. Only an unbiased manifestation is observable. The method proposed, provides an adjustment to obtain unbiased estimates of model parameters. The correction of bias, however, is not possible without additional identifying information. An instrumental variable is a practical form of additional information that can be used for this purpose. By treating the unobservable explanatory variable as 'missing' data the Markov Chain Monte Carlo Expectation Maximization (MCEM) algorithm is applied for maximum likelihood estimation of the parameters of a measurement error model with identifying information in the form of an instrumental variable. Implementation strategies, computational aspects, behavior of the estimators and inference resulting from application of the MCEM algorithm to the instrumental variable measurement error model are studied. A general methodology is developed that encompasses a variety of previously studied special case models and it is shown how they all can be modeled and estimated using the MCEM algorithm. Through our method it is shown how a structural logistic regression measurement error model can be directly fitted without the probit approximation. This was not possible prior to the research presented in this dissertation. The proposed methodology is compared numerically with the exact maximum likelihood estimates for two normal family models. Also, the behavior of the method is investigated when one of the variance parameters is near the boundary of the parameter space. The problem of measurement error in a survival time model with right censoring is considered and it is shown how the proposed method can be used to estimate a hazard function model, by construction of some special likelihoods and further methodological development. Two methods have been proposed, one of which is a semi-parametric method and the other is full parametric.