MM Algorithms for Some Discrete Multivariate Distributions.

Area of Science:

• Statistics
• Optimization
• Computational Statistics

Background:

• The minorization-maximization (MM) principle is a powerful framework for developing optimization algorithms.
• While Expectation-Maximization (EM) algorithms are a subset of MM algorithms, not all MM algorithms are EM algorithms.

Purpose of the Study:

• To derive and present MM algorithms for maximum likelihood estimation (MLE) in various discrete multivariate distributions.
• To highlight the applicability of MM algorithms to high-dimensional statistical problems.

Main Methods:

• Derivation of MM algorithms tailored for specific discrete multivariate distributions.
• Comparison of MM algorithm performance against Newton's method for MLE.
• Application to the classification of handwritten digits using high-dimensional data.

Main Results:

• Successfully derived MM algorithms for Dirichlet-multinomial, Connor-Mosimann, Neerchal-Morel, negative-multinomial, and zero-inflated/truncated distributions.
• MM algorithms demonstrated reliable convergence to maximum likelihood estimates.
• MM algorithms showed particular relevance for high-dimensional problems due to the absence of matrix inversion.

Conclusions:

• MM algorithms provide an effective and computationally efficient alternative to traditional methods like Newton's for MLE.
• The MM principle is a versatile tool applicable to a wide range of statistical modeling and estimation tasks.