A polyhedral method for solving sparse polynomial systems
Mathematics of Computation
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Solving algebraic equations in terms of A-hypergeometric series
Discrete Mathematics
Parameter estimation in linear models with heteroscedastic variances subject to order restrictions
Journal of Multivariate Analysis
Genetic algorithms and maximum likelihood estimation
Genetic algorithms and maximum likelihood estimation
A Survey of Statistical Network Models
Foundations and Trends® in Machine Learning
The multivariate Behrens-Fisher distribution
Journal of Multivariate Analysis
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In statistical inference, mixture models consisting of several component subpopulations are used widely to model data drawn from heterogeneous sources. In this paper, we consider maximum likelihood estimation for mixture models in which the only unknown parameters are the component proportions. By applying the theory of multivariable polynomial equations, we derive bounds for the number of isolated roots of the corresponding system of likelihood equations. If the component densities belong to certain familiar continuous exponential families, including the multivariate normal or gamma distributions, then our upper bound is, almost surely, the exact number of solutions.