Improving the vector $$\varepsilon $$ε acceleration for the EM algorithm using a re-starting procedure

The expectation–maximization (EM) algorithm is a popular algorithm for finding maximum likelihood estimates from incomplete data. However, the EM algorithm converges slowly when the proportion of missing data is large. Although many acceleration algorithms have been proposed, they require complex calculations. Kuroda and Sakakihara (Comput Stat Data Anal 51:1549–1561, 2006) developed the $$\varepsilon $$ε-accelerated EM algorithm which only uses the sequence of estimates obtained by the EM algorithm to get an accelerated sequence for the EM sequence but does not change the original EM sequence. We find that the accelerated sequence often has larger values of the likelihood than the current estimate obtained by the EM algorithm. Thus, in this paper, we try to re-start the EM iterations using the accelerated sequence and then generate a new EM sequence that increases its speed of convergence. This algorithm has another advantage of simple implementation since it only uses the EM iterations and re-starts the iterations by an estimate with a larger likelihood. The re-starting algorithm called the $$\varepsilon $$εR-accelerated EM algorithm can further improve the EM algorithm and the $$\varepsilon $$ε-accelerated EM algorithm in the sense of that it can reduces the number of iterations and computation time.