On the global and componentwise rates of convergence of the EM algorithm

Abstract

The EM algorithm is a very general and popular iterative algorithm in statistics for finding maximum-likelihood estimates in the presence of incomplete data. In the paper that defined and popularized EM, Dempster, Laird, and Rubin (1977) showed that its global rate of convergence is governed by the largest eigenvalue of the matrix of fractions of missing information due to incomplete data. It was also mentioned that componentwise rates of convergence can differ from each other when the fractions of information loss vary across different components of a parameter vector. In this article, using the well-known diagonability theorem, we present a general description on how and when the componentwise rates differ, as well as their relationships with the global rate. We also provide an example, a standard contaminated normal model, to show that such phenomena are not necessarily pathological, but can occur in useful statistical models.