We study the distribution of the ratio of two central Wishart matrices with different covariance matrices. We first derive the density function of a particular matrix form of the ratio and show that its cumulative distribution function can be express
ed in terms of the hypergeometric function 2F1 of a matrix argument. Then we apply the holonomic gradient method for numerical evaluation of the hypergeometric function. This approach enables us to compute the power function of Roys maximum root test for testing the equality of two covariance matrices.
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 to solve a conditional maximum likelihood estimation problem in statistics. Our interplay between the theory of hypergeometric functions and statistics gives some new formulas of $A$-hypergeometric polynomials.
We give two efficient methods to derive Pfaffian systems for A-hypergeometric systems for the application to the holonomic gradient method for statistics. We utilize the Hilbert driven Buchberger algorithm and Macaulay type matrices in the two methods.
This is the third revision. We study bases of Pfaffian systems for $A$-hypergeometric system. Grobner deformations give bases. These bases also give those for twisted cohomology groups. For hypergeometric system associated to a class of order polytop
es, these bases have a combinatorial description. The size of the bases associated to a subclass of the order polytopes have the growth rate of the polynomial order. Bases associated to two chain posets and bouquets are studied.
We give a new algorithm to find local maximum and minimum of a holonomic function and apply it for the Fisher-Bingham integral on the sphere $S^n$, which is used in the directional statistics. The method utilizes the theory and algorithms of holonomic systems.
We give algorithms of computing bases of logarithmic cohomology groups for square-free polynomials in two variables. (Fixed typos of v1)
We will introduce a modified system of A-hypergeometric system (GKZ system) by applying a change of variables for Groebner deformations and study its Groebner basis and the indicial polynomials along the exceptional hypersurface.
We present an algorithm to construct a basis of k-th extension group of a D-module M in ring of the formal power series Ext_D^k(M,O).
We describe algorithms for computing various functors for algebraic D-modules, i.e. systems of linear partial differential equations with polynomial coefficients. We will give algorithms for restriction, tensor product, localization, and local cohomology groups for all degrees.
We give an algorithm to compute the following cohomology groups on $U = C^n setminus V(f)$ for any non-zero polynomial $f in Q[x_1, ..., x_n]$; 1. $H^k(U, C_U)$, $C_U$ is the constant sheaf on $U$ with stalk $C$. 2. $H^k(U, Vsc)$, $Vsc$ is a locally
constant sheaf of rank 1 on $U$. We also give partial results on computation of cohomology groups on $U$ for a locally constant sheaf of general rank and on computation of $H^k(C^n setminus Z, C)$ where $Z$ is a general algebraic set. Our algorithm is based on computations of Grobner bases in the ring of differential operators with polynomial coefficients.
