### Abstract:

Factor analysis is an important statistical tool used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables which are called factors. Maximum likelihood estimation (MLE) has been popular for fitting factor analysis. Among variety of iterative methods that can be used to perform MLE, the EM algorithm is probably one of the most stable in terms of monotonely increasing the likelihood and the easiest to implement. However, in the real world, the rate of convergence of EM could be painfully slow in factor model estimation. In this dissertation, we study two popular problems in algorithms and structures for covariance estimates. The first problem is factor analysis and mixture of factor analyzers models estimation by using the alpha-EM algorithm. In the alpha-EM algorithm we replace the logarithm by alpha-logarithm. Logarithms have important roles besides the derivation of the log-EM algorithm.The Kullback-Leibler divergence and Fisher information matrix all bring about the logarithm. For alpha-logarithm with different values of alpha we actually have other important information measurements such as the Hellinger distance and weighted square distance besides the Kullback-Leibler divergence. After calculation we get two non-tractable update equations in alpha-EM. In order to get tractable update equations as we have in log-EM, we need to do two more things. One of them is iteration index shifting and the other one is series expansion. These two steps are necessary for practical reasons. In addition, we apply the alpha-EM algorithm to actual financial data. The speed of convergence is much faster than traditional log-EM algorithm and you could choose different values of alpha to achieve the best rate of convergence. The second problem is covariance estimation by using matrix fraction representations. There is a vast literature that suggests factor models for dealing with covariance estimation. One of the important reason is that we can interpret the statistical factors by actual financial indicators. Here, we consider using matrix fraction representations. One of the many reasons that this would be a better idea than factor model is that the inverse of a factor model no longer have the same factor structure. But fraction representations don't have this problem. Another reason is that factor model is not a convex set. But band fraction representation is a convex set. More importantly we can show that factor model is a special case of band fraction representation. That means if the covariance matrices have factor structure we still use band fraction representation. It had been expected that band fraction representation would be better than factor model. We show the foresight is true.