Holonomic gradient method for the distribution function of the largest root of a Wishart matrix

We apply the holonomic gradient method introduced by Nakayama et al.(2011) to the evaluation of the exact distribution function of the largest root of a Wishart matrix, which involves a hypergeometric function 1F1 of a matrix argument. Numerical evaluation of the hypergeometric function has been one of the longstanding problems in multivariate distribution theory. The holonomic gradient method offers a totally new approach, which is complementary to the infinite series expansion around the origin in terms of zonal polynomials. It allows us to move away from the origin by the use of partial differential equations satisfied by the hypergeometric function. From numerical viewpoint we show that the method works well up to dimension 10. From theoretical viewpoint the method offers many challenging problems both to statistics and D-module theory

Computation of the Expected Euler Characteristic for the Largest Eigenvalue of a Real Non-central Wishart Matrix

We give an approximate formula for the distribution of the largest eigenvalue of real Wishart matrices by the expected Euler characteristic method for the general dimension. The formula is expressed in terms of a definite integral with parameters. We derive a differential equation satisfied by the integral for the $2 \times 2$ matrix case and perform a numerical analysis of it

Holonomic Decent Minimization Method for Restricted Maximum Likelihood Estimation

Recently, the school of Takemura and Takayama have developed a quite interesting minimization method called holonomic gradient descent method (HGD). It works by a mixed use of Pfaffian differential equation satisfied by an objective holonomic function and an iterative optimization method. They successfully applied the method to several maximum likelihood estimation (MLE) problems, which have been intractable in the past. On the other hand, in statistical models, it is not rare that parameters are constrained and therefore the MLE with constraints has been surely one of fundamental topics in statistics. In this paper we develop HGD with constraints for MLE

Holonomic Gradient Method-Based CDF Evaluation for the Largest Eigenvalue of a Complex Noncentral Wishart Matrix

The outage probability of maximal-ratio combining (MRC) for a multiple-input multiple-output (MIMO) wireless communications system under Rician fading is given by the cumulative distribution function (CDF) for the largest eigenvalue of a complex noncentral Wishart matrix. This CDF has previously been expressed as a determinant whose elements are integrals of a confluent hypergeometric function. For the determinant elements, conventional evaluation approaches, e.g., truncation of infinite series ensuing from the hypergeometric function or numerical integration, can be unreliable and slow even for moderate antenna numbers and Rician $K$-factor values. Therefore, herein, we derive by hand and by computer algebra also differential equations that are then solved from initial conditions computed by conventional approaches. This is the holonomic gradient method (HGM). Previous HGM-based evaluations of MIMO relied on differential equations that were not theoretically guaranteed to converge, and, thus, yielded reliable results only for few antennas or moderate $K$. Herein, we reveal that gauge transformations can yield differential equations that are {\emph{stabile}}, i.e., guarantee HGM convergence. The ensuing HGM-based CDF evaluation is demonstrated reliable, accurate, and expeditious in computing the MRC outage probability even for very large antenna numbers and values of $K$

Calculation of orthant probabilities by the holonomic gradient method

We apply the holonomic gradient method (HGM) introduced by [9] to the calculation of orthant probabilities of multivariate normal distribution. The holonomic gradient method applied to orthant probabilities is found to be a variant of Plackett's recurrence relation ([14]). However an implementation of the method yields recurrence relations more suitable for numerical computation than Plackett's recurrence relation. We derive some theoretical results on the holonomic system for the orthant probabilities. These results show that multivariate normal orthant probabilities possess some remarkable properties from the viewpoint of holonomic systems. Finally we show that numerical performance of our method is comparable or superior compared to existing methods.Comment: 17 page

Calculation and Properties of Zonal Polynomials

We investigate the zonal polynomials, a family of symmetric polynomials that appear in many mathematical contexts, such as multivariate statistics, differential geometry, representation theory, and combinatorics. We present two computer algebra packages, in SageMath and in Mathematica, for their computation. With the help of these software packages, we carry out an experimental mathematics study of some properties of zonal polynomials. Moreover, we derive and prove closed forms for several infinite families of zonal polynomial coefficients

Estimation of exponential-polynomial distribution by holonomic gradient descent

We study holonomic gradient decent for maximum likelihood estimation of exponential-polynomial distribution, whose density is the exponential function of a polynomial in the random variable. We first consider the case that the support of the distribution is the set of positive reals. We show that the maximum likelihood estimate (MLE) can be easily computed by the holonomic gradient descent, even though the normalizing constant of this family does not have a closed-form expression and discuss determination of the degree of the polynomial based on the score test statistic. Then we present extensions to the whole real line and to the bivariate distribution on the positive orthant

Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix

This paper discusses certain properties of heterogeneous hypergeometric functions with two matrix arguments. These functions are newly defined but have already appeared in statistical literature and are useful when dealing with the derivation of certain distributions for the eigenvalues of singular beta-Wishart matrices. The joint density function of the eigenvalues and the distribution of the largest eigenvalue can be expressed in terms of certain heterogeneous hypergeometric functions. Exact computation of the distribution of the largest eigenvalue is conducted here for a real case

The Annihilating Ideal of the Fisher Integral

In this paper, we discuss a system of differential equations for the Fisher integral on the special orthogonal group. Especially, we explicitly give a set of linear differential operators which generates the annihilating ideal of the Fisher integral, and we prove that the annihilating ideal is a maximal left ideal of the ring of differential operators with polynomial coefficients. Our proof is given by a discussion concerned with an annihilating ideal of a Schwartz distribution associated with the Haar measure on the special orthogonal group. We also give differential operators annihilating the Fisher integral for the diagonal matrix by a new approach.Comment: 11 page

Expressing the largest eigenvalue of a singular beta F-matrix with heterogeneous hypergeometric functions

In this paper, the exact distribution of the largest eigenvalue of a singular random matrix for multivariate analysis of variance (MANOVA) is discussed. The key to developing the distribution theory of eigenvalues of a singular random matrix is to use heterogeneous hypergeometric functions with two matrix arguments. In this study, we define the singular beta F-matrix and extend the distributions of a nonsingular beta F -matrix to the singular case. We also give the joint density of eigenvalues and the exact distribution of the largest eigenvalue in terms of heterogeneous hypergeometric functions.Comment: The title is changed (the old title is "The exact distribution of the largest eigenvalue of a singular beta F-matrix for Roy's test"