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Maximum likelihood for matrices with rank constraints. (English) Zbl 1345.62043
Summary: Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.

MSC:
62F10 Point estimation
13P15 Solving polynomial systems; resultants
14M12 Determinantal varieties
62F30 Parametric inference under constraints
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
90C90 Applications of mathematical programming
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