MM Algorithms for Variance Components Models

Hua Zhou, Liuyi Hu, Jin Zhou, Kenneth Lange

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


Variance components estimation and mixed model analysis are central themes in statistics with applications in numerous scientific disciplines. Despite the best efforts of generations of statisticians and numerical analysts, maximum likelihood estimation (MLE) and restricted MLE of variance component models remain numerically challenging. Building on the minorization–maximization (MM) principle, this article presents a novel iterative algorithm for variance components estimation. Our MM algorithm is trivial to implement and competitive on large data problems. The algorithm readily extends to more complicated problems such as linear mixed models, multivariate response models possibly with missing data, maximum a posteriori estimation, and penalized estimation. We establish the global convergence of the MM algorithm to a Karush–Kuhn–Tucker point and demonstrate, both numerically and theoretically, that it converges faster than the classical EM algorithm when the number of variance components is greater than two and all covariance matrices are positive definite. Supplementary materials for this article are available online.

Original languageEnglish (US)
Pages (from-to)350-361
Number of pages12
JournalJournal of Computational and Graphical Statistics
Issue number2
StatePublished - Apr 3 2019


  • Global convergence
  • Linear mixed model (LMM)
  • Matrix convexity
  • Maximum a posteriori (MAP) estimation
  • Minorization–maximization (MM)
  • Multivariate response
  • Penalized estimation
  • Variance components model

ASJC Scopus subject areas

  • Statistics and Probability
  • Discrete Mathematics and Combinatorics
  • Statistics, Probability and Uncertainty


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