Abstract
A distribution whose normalization constant is an A-hypergeometric polynomial is called an A-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an A-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associated with the rational normal curve, appears in inferences involving exchangeable partition structures. An exact sampling algorithm is presented for the general (any number of rows) A-hypergeometric distributions. Then, the maximum likelihood estimation of the A-hypergeometric distribution associated with the rational normal curve, which is an algebraic exponential family, is discussed. The information geometry of the Newton polytope is useful for analyzing the full and the curved exponential family. Algebraic methods are provided for evaluating the A-hypergeometric polynomials.
Highlights
The A-hypergeometric function introduced by Gel’fand, Kapranov, and Zelevinsky [1] is a solution of the A-hypergeometric system of partial differential equations
Takayama et al [2] called a distribution whose normalization constant is an A-hypergeometric polynomial as an A-hypergeometric distribution
If the system is associated with the rational normal curve, equivalently, if im−1 = m − 1 in (2.10), the A-hypergeometric polynomial (2.8) is a constant multiple of the associated partial Bell polynomial defined by Definition 2.1
Summary
The A-hypergeometric function introduced by Gel’fand, Kapranov, and Zelevinsky [1] is a solution of the A-hypergeometric system of partial differential equations. Gradient-based methods to evaluate the MLE will be discussed They are demonstrated in a problem associated with an EPPF that appears in an empirical Bayes approach. All the above applications demand practical methods for evaluating the A-hypergeometric polynomials associated with the rational normal curve. By virtue of this explicit expression, alternative algebraic methods for evaluating the A-hypergeometric polynomials are presented. They are examples of methods called the holonomic gradient methods (HGMs) [11, 12, 13]. The difference HGM demands less computational cost, while the recurrence relation gives more accurate estimates The performance of these methods are compared in applications to evaluating specific A-hypergeometric polynomials. The accuracy of known asymptotic form and that obtained by the method developed by Takayama et al [2] are compared
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