On the orderings and groupings of conditional maximizations within ECM-type algorithms

The ECM and ECME algorithms are generalizations of the EM algorithm in which the maximization (M) step is replaced by several conditional maximization (CM)I steps, The order that the Chi-steps are performed is trivial to change and generally affects how fast the algorithm converges. Moreover, the same order of Chi-steps need not be used at each iteration and in some applications it is feasible to group two or more CM-steps into one larger CM-step. These issues also arise when implementing the Gibbs sampler, and in this article we study them in the context of fitting log-linear and random-effects models with ECM-type algorithms. We find that some standard theoretical measures of the tate of convergence fan be of Little use in comparing the computational time required, and that common strategies such as using a random ordering may not provide the desired effects. We also develop two algorithms for fitting random-effects models to illustrate that with careful selection of CM-steps, ECM-type algorithms can be substantially faster than the standard EM algorithm.