On Computing the Largest Fraction of Missing Information for the EM Algorithm and the Worst Linear Function for Data Augmentation

We address the problem of computing the largest fraction of missing information for the EM algorithm and the worst linear function for data augmentation These are the largest eigenvalue and its associated eigenvector for the Jacobian of the EM operator at a maximum likelihood estimate which are important for assessing convergence in iterative simulation An estimate of the largest fraction of missing information is available from the EM iterates this is often adequate since only a few gures of accuracy are needed In some instances the EM iteration also gives an estimate of the worst linear function We show that improved estimates can be essential for proper inference In order to obtain improved estimates e ciently we use the power method for eigen computation Unlike eigenvalue decomposition the power method computes only the largest eigenvalue and eigenvector of a matrix it can take advantage of a good eigenvector estimate as an initial value and it can be terminated after only a few gures of accuracy are achieved Moreover the matrix products needed in the power method can be computed by extrapolation obviating the need to form the Jacobian of the EM operator We give results of simulation studies on multivariate normal data showing that this approach becomes more e cient as the data dimension increases than methods that use a nite di erence approximation to the Jacobian which is the only general purpose alternative available