Counting and locating the solutions of polynomial systems of maximum likelihood equations, I
Journal of Symbolic Computation
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In this thesis we apply simple genetic algorithms to solve constrained maximum likelihood estimation problems which are analytically intractable and lack closed-form solutions. That is, a genetic algorithm is used in three statistical optimization problems which are generally solved by numerical methods. The first problem involves the estimation of component mixture weights in a finite mixture model with known component density functions. The goal of the second problem is to determine an estimate of a proportion parameter when given selected binomial information. In the third problem, we seek to estimate a matrix parameter of a Wishart likelihood subject to Loewner order constraints. The results of these computational experiments illustrate that the genetic algorithm approach performs well in comparison to existing methods. To complement the simulation work, we introduce a general convergence rate result which is derived by the application of iterated function theory. In addition, by examining the likelihood functions described above from an algebraic perspective, we obtain some useful insights; specifically, we demonstrate the equivalence of maximum likelihood estimation and polynomial root finding in the mixture model estimation problem presented. We apply results in polynomial theory (e.g., Descartes' Rule of Signs, Budan-Fourier Theorem, Sturm's Theorem) to the estimation of component mixing weights, and introduce a genetic algorithm approach for locating the roots of a system of polynomials. For mixture models in which the component density functions are completely specified, we derive a bound on the number of complex solutions to the system of multivariate polynomials corresponding to the score equations. A crucial step in the proof of the bound is the application of Bernstein's Theorem to the system of score equations associated with the mixture model estimation problem.