Estimação de máxima verossimilhança via algoritmo EM

Abstract

This paper aims to present the EM algorithm, which is a software tool used for calculating the maximum likelihood estimator (MLE) in an iterative manner, especially in problems involving missing data. For this we need to get the complete data set which is the set of observed data augmented with the set of missing data, from there get the log-likelihood function associated with complete data. We know that this algorithm converges surely to MLE and is based on the idea of replacing a difficult maximization by a sequence of maximizations easier, involving two steps, the step “E”(Expectation) calculates the expected value of the complete log likelihood, and the step “M”(Maximization), which finds its fullest. The steps are repeated until convergence is achieved. Before exemplify the calculation of MLE via EM algorithm, define the hier- archical models that have the advantage of modeling complicated processes through a sequence of relatively simple models, placed in a hierarchy. In addition, dealing with the hierarchy is no more difficult to deal with marginal or conditional distributions. As an example of calculating MLE via EM algorithm, we use regression models, where the errors take on student-t distribution, which in addition to being symmetrical, have heavier tails (robust to accommodate extreme values). We will also present applica- tions for these distribution.