General A Tutorial on MM Algorithms

Most problems in frequentist statistics involve optimization of a function such as a likelihood or a sum of squares. EM algo-rithms are among the most effective algorithms for maximum likelihood estimation because they consistently drive the likeli-hood uphill by maximizing a simple surrogate function for the log-likelihood. Iterative optimization of a surrogate function as exempli ed by an EM algorithm does not necessarily require missing data. Indeed, every EM algorithm is a special case of the more general class of MM optimization algorithms, which typically exploit convexity rather than missing data in majoriz-ing or minorizing an objective function. In our opinion, MM algorithms deserve to be part of the standard toolkit of profes-sional statisticians. This article explains the principle behind MM algorithms, suggests some methods for constructing them, and discusses some of their attractive features. We include nu-merous examples throughout the article to illustrate the concepts described. In addition to surveying previous work on MM algo-rithms, this article introduces some newmaterial on constrained optimization and standard error estimation