A-hypergeometric distributions and Newton polytopes

We give a bijection between a quotient space of the parameters and the space of moments for any A-hypergeometric distribution. An algorithmic method to compute the inverse image of the map is proposed utilizing the holonomic gradient method and an asymptotic equivalence of the map and the iterative proportional scaling. The algorithm gives a method for solving a conditional maximum likelihood estimation problem in statistics. The interplay between the theory of hypergeometric functions and statistics allows us to give some new formulas for A-hypergeometric polynomials.