Hypergeometric Functions Of Matrix Argument Research Articles

Laplace approximation for Bessel functions of matrix argument

We derive Laplace approximations to three functions of matrix argument which arise in statistics and elsewhere: matrix Bessel A ν ; matrix Bessel B ν ; and the type II confluent hypergeometric function of matrix argument, Ψ. We examine the theoretical and numerical properties of the approximations. On the theoretical side, it is shown that the Laplace approximations to A ν , B ν and Ψ given here, together with the Laplace approximations to the matrix argument functions 1 F 1 and 2 F 1 presented in Butler and Wood (Laplace approximations to hyper-geometric functions with matrix argument, Ann. Statist. (2002)), satisfy all the important confluence relations and symmetry relations enjoyed by the original functions.

Complex random matrices and Rayleigh channel capacity

The eigenvalue densities of complex central Wishart matrices are investigated with the objective of studying an open problem in channel capacity. These densities are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. The connection between the complex Wishart matrix theory and information theory is given. This facilitates the evaluation of the most important information-theoretic measure, the so-called channel capacity. In particular, the capacity of multiple input, multiple output (MIMO) Rayleigh distributed channels are fully investigated. We consider both correlated and uncorrelated channels and derive the corresponding channel capacity formulas. It is shown how the channel correlation degrades the capacity of the communication system. To appear in Communications in Information & Systems

Constant term identities and hypergeometric functions on spaces of Hermitian matrices

We announce the generalization, to the setting of hypergeometric functions of matrix argument, of classical constant term identities. As special cases, we obtain some constant term identities which play a prominent role in the proof of Dyson's conjecture.

The distribution and moments of the smallest eigenvalue of a random matrix of wishart type

Given a random rectangular m × n matrix with elements from a normal distribution, what is the distribution of the smallest singular value? To pose an equivalent question in the language of multivariate statistics, what is the distribution of the smallest eigenvalue of a matrix from the central Wishart distribution in the null case? We give new results giving the distribution as a simple recursion. This includes the more difficult case when n – m is an even integer, without resorting to zonal polynomials and hypergeometric functions of matrix arguments. With the recursion, one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials, exponentials, and at worst ordinary hypergeometric functions. We further elaborate on the special cases when n – m = 0, 1, 2, and 3 and give a numerical table of the expected values for 2 ⩽ m ⩽ 25 and 0 ⩽ n – m ⩽ 25.

Hypergeometric functions on complex matrix space

The authors had presented the general foundations for a theory of hypergeometric functions of matrix argument over real division algebras. Here, they further develop the fine structure of these functions over the complex field, including series expansions, integral representations, asymptotics, differential equations, addition formulas, multiplication formulas, summation theorems, transformation properties, etc. Especially important are the operator-valued hypergeometric functions, required for (nonspherical) expansions such as addition formulas by the noncommutativity of matrix multiplication

Totally positive kernels, pólya frequency functions, and generalized hypergeometric series

Recently, K. I. Gross and the author [ J. Approx. Theory 59:224–246 (1989)] analyzed the total positivity of kernels of the form K(x,y) = p F q(xy) , x, y ϵ R, where p F q denotes a classical generalized hypergeometric series. There, we related the determinants which define the total positivity of K to the hypergeometric functions of matrix argument, defined on the space of n × n Hermitian matrices. In the first part of this paper, we apply these methods to derive the total positivity properties of the kernels K for more general choices of the parameters in these hypergeometric series. In particular, we prove that if a i > 0 and k i is a positive integer ( i = 1,…, p) then K(x,y) = p F q(a 1,…,a p;a 1 + k 1,…,a p + k p; xy) is strictly totally positive on R 2. In the second part, we use the theory of entire functions to derive some Pólya frequency function properties of the hypergeometric series p F q ( x). Some of these latter results suggest a curious duality with the total positivity properties described above. For instance, we prove that if p > 1 and the a i and k i are as above, then there exists a probability density function f on R, such that f is a strict Pólya frequency function, and 1/ p F p(a 1+k 1,…,a p+k p;a 1,…,a p;z= Lf(z) , the Laplace transform of f.

Total positivity, spherical series, and hypergeometric functions of matrix argument

Given a totally positive function K of two real variables, is there a method for establishing the total positivity of K in an “obvious” fashion? In the case in which K( x, y) = f( xy), where f is real-analytic in a neighborhood of zero, we obtain integral representations for the determinants which define the total positivity of K. The total positivity of K then follows immediately from positivity of the integrands. In particular, we analyze the total positivity of classical hypergeometric functions by these methods. The central theme of this work is the circle of ideas that relates total positivity to “spherical series” on the symmetric space GL(n, C) U(n) , and classical hypergeometric functions to hypergeometric functions of matrix argument.

Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions

Hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis, multivariate statistics, quantum physics, molecular chemistry, and number theory. This paper presents a general theory of such functions for real division algebras. These functions, which generalize the classical hypergeometric functions, are defined by infinite series on the space S = S ( n , F ) S = S(n,\,\mathbf {F}) of all n × n n \times n Hermitian matrices over the division algebra F \mathbf {F} . The theory depends intrinsically upon the representation theory of the general linear group G = G L ( n , F ) G = GL(n,\,\mathbf {F}) of invertible n × n n \times n matrices over F \mathbf {F} , and the theme of this work is the full exploitation of the inherent group theory. The main technique is the use of the method of “algebraic induction” to realize explicitly the appropriate representations of G G , to decompose the space of polynomial functions on S S , and to describe the “zonal polynomials” from which the hypergeometric functions are constructed. Detailed descriptions of the convergence properties of the series expansions are given, and integral representations are provided. Future papers in this series will develop the fine structure of these functions.

Generalized Laplace transform with matrix variables

In the present paper we have extended generalized Laplace transforms of Joshi to the space of m × m symmetric matrices using the confluent hypergeometric function of matrix argument defined by Herz as kernel. Our extension is given by urn:x-wiley:01611712:media:ijmm586307:ijmm586307-math-0001 The convergence of this integral under various conditions has also been discussed. The real and complex inversion theorems for the transform have been proved and it has also been established that Hankel transform of functions of matrix argument are limiting cases of the generalized Laplace transforms.

Series Expansion for a Hypergeometric Function of Matrix Argument with Applications

SummaryA series expansion is obtained for the confluent hypergeometric function of the second kind when the argument is a 2 times 2 positive definite matrix. Applications are made to the distributions of Hotelling's generalized T02 statistic, and the smallest latent root of the covariance matrix.

On umvu estimators for the multivariate lognormal distribution and their variances

Uniformly minimum variance unbiased estimators of several parameters of the multivariate lognormal distribution are expressed by using the hypergeometric functions of matrix argument. And the variances are given in special cases.

Latent Roots and Matrix Variates: A Review of Some Asymptotic Results

The exact noncentral distributions of matrix variates and latent roots derived from normal samples involve hypergeometric functions of matrix argument. These functions can be defined as power series, by integral representations, or as solutions of differential equations, and there is no doubt that these mathematical characterizations have been a unifying influence in multivariate noncentral distribution theory, at least from an analytic point of view. From a computational and inference point of view, however, the hypergeometric functions are themselves of very limited value due primarily to the many difficulties involved in evaluating them numerically and consequently in studying the effects of population parameters on the distributions. Asymptotic results for large sample sizes or large population latent roots have so far proved to be much more useful for such problems. The purpose of this paper is to review some of the recent results obtained in these areas.

Expressions for some hypergeometric functions of matrix argument with applications

Reasonably simple expressions are given for some hypergeometric functions when the size of the argument matrix or matrices is two. Applications of these expressions in connection with the distributions of the latent roots of a 2 × 2 Wishart matrix are also given.

Systems of Partial Differential Equations for Hypergeometric Functions of Matrix Argument

Many distributions in multivariate analysis can be expressed in a form involving hypergeometric functions $_pF_q$ of matrix argument e.g. the noncentral Wishart $(_0F_1)$ and the noncentral multivariate $F(_1F_1)$. For an exposition of distributions in this form see James [9]. The hypergeometric function $_pF_q$ has been defined by Constantine [1] as the power series representation \begin{equation*}\tag{1.1} _pF_q(a_1,\cdots, a_p; b_1,\cdots, b_q; R) = \sum^\infty_{k=0} \sum_\kappa \frac{(a_1)_\kappa\cdots(a_p)_\kappa}{(b_1)_\kappa\cdots (b_q)_\kappa} \frac{C_\kappa (R)}{k!}\end{equation*} where $a_1,\cdots, a_p, b_1,\cdots, b_q$ are real or complex constants, $(a)_\kappa = \mathbf{prod}^m_{i=1}(a - \frac{1}{2}(i - 1))_{k_i},\quad (a)_n = a(a + 1)\cdots (a + n - 1)$ and $C_\kappa(R)$ is the zonal polynomial of the $m \times m$ symmetric matrix $R$ corresponding to the partition $\kappa = (k_1, k_2,\cdots, k_m), k_1 \geqq k_2 \geqq \cdots \geqq k_m$, of the integer $k$ into not more than $m$ parts. The functions defined by (1.1) are identical with the hypergeometric functions defined by Herz [5] by means of Laplace and inverse Laplace transforms. For a detailed discussion of hypergeometric functions and zonal polynomials, the reader is referred to the papers [1] of Constantine and [7], [8], [9] of James. From a practical point of view, however, the series (1.1) may not be of great value. Although computer programs have been developed for calculating zonal polynomials up to quite high order, the series (1.1) may converge very slowly. It appears that some asymptotic expansions for such functions must be obtained. It is well known that asymptotic expansions for a function can in many cases be derived using a differential equation satisfied by the function (see e.g. Erdelyi [4]), and so, with this in mind, a study of differential equations satisfied by certain hypergeometric functions certainly seems justified. In this paper a conjecture due to A. G. Constantine is verified i.e. it is shown that the function \begin{equation*}\tag{1.2} _2F_1(a, b; c; R) = \sum^\infty_{k=0} \sum_\kappa \frac{(a)_\kappa(b)_\kappa}{(c)_\kappa} \frac{C_\kappa(R)}{K!}\end{equation*} satisfies the system of partial differential equations \begin{align*} \tag{1.3} R_i(1 &- R_i)\partial^2F/\partial R_i^2 + \{ c - \frac{1}{2}(m - 1) - (a + b + 1 - \frac{1}{2}(m - 1))R_i \\ &+\frac{1}{2} \sum^m_{j=1,j\neq i}\lbrack R_i(1 - R_i)/(R_i - R_j) \rbrack\}\partial F/\partial R_i \\ &-\frac{1}{2} \sum^m_{j=1,j\neq i} \lbrack R_j(1 - R_j)/(R_i - R_j) \rbrack\partial F/\partial R_j = abF \quad (i = 1,2, \cdots, m)\end{align*} where $R_1, R_2,\cdots, R_m$ are the latent roots of the complex symmetric $m \times m$ matrix $R$. When $m = 1$, the system (1.3) clearly reduces to the classical hypergeometric equation. It appears difficult to establish this conjecture directly, and the method used has necessitated a section devoted to a summary of the argument involved (Section 3). The main result in the paper is summarized in Theorem 3.1 of this section. Section 4 contains proofs referred to in Section 3. Using the fact that $C_\kappa (R)$ satisfies the partial differential equation (James [10])\begin{equation*}\tag{1.4} \sum^m_{i=1} R_i^2\partial^2y/\partial R_i^2 + \sum^m_{j=1,j\neq i}\lbrack R_i^2/(R_i - R_j) \rbrack\partial y/\partial R_i = \sum^m_{i=1}k_i(k_i + m - i - 1)y,\end{equation*} James and Constantine [11] have obtained the effects of certain differential operators on $C_\kappa(R)$. These results are given in Section 2 and are used in many proofs in Section 4. Section 5 is probably of most interest statistically, for here systems of partial differential equations similar to (1.3) are given for $_1F_1(a; c; R)$ and $_0F_1(c; R)$. These two functions occur often in multivariate distributions. The differential equations for $_1F_1(a; c; R)$ have been used by Constantine [3] to obtain an asymptotic expansion for the noncentral likelihood ratio criterion, and by the author [12] to obtain asymptotic distributions of Hotelling's generalized $T_0^2$ statistic, Pillai's $V^{(m)}$ criterion, and for the largest latent root of the covariance matrix. The system for $_0F_1(c; R)$ is a generalization of that given by James [6] for $_0F_1(m/2; R)$.

Asymptotic Distributions of Some Multivariate Tests

In this paper it is shown how the systems of partial differential equations developed by the author [15] for the hypergeometric functions of matrix argument may be used to obtain asymptotic expansions for some distributions occurring in multivariate analysis. In particular expansions are derived for the distributions of Hotelling's generalized $T_0^2$-statistic, Pillai's $V^{(s)}$ criterion, and the largest latent root of the sample covariance matrix.

A System of Linear Differential Equations for the Distribution of Hotelling's Generalized $T_o^2$

Let $\mathscr{S}_1, \mathscr{S}_2$ be independent $m \times m$ matrices on $n_1, n_2$ degrees of freedom respectively, $\mathscr{S}_2$ having a Wishart distribution and $\mathscr{S}_1$ having a possibly non-central Wishart distribution with the same covariance matrix. Hotelling's generalized $T_0^2$ statistic is then defined [7] by \begin{equation*}\tag{1.1}T = n^{-1}_2T_0^2 = \operatorname{tr} \mathscr{S}_1\mathscr{S}^{-1}_2.\end{equation*} The complete distribution of this statistic is known only in particular cases. If $m = 1,$ then $(n_2/n_1)T$ is simply non-central $F$. In the case $n_1 = 1, T$ reduces to Hotelling's generalization of "Student's" $t,$ which also has a non-central $F$ distribution. When $m = 2$, Hotelling [7] has shown that in the null case the density function of $T$ is \begin{equation*}\tag{1.2}f(T) = \lbrack\Gamma(n_1 + n_2 - 1)/\Gamma(n_1)\Gamma (n_2 - 1)\rbrack(\frac{1}{2}T)^{n_1-1}(1 + \frac{1}{2}T)^{-(n_1+n_2)} \cdot_2F_1 (1, \frac{1}{2}(n_1 + n_2); \frac{1}{2}(n_2 + 1); v),\end{equation*} where $\nu = T^2/(T + 2)^2$, and $_2F_1$ is the Gaussian hypergeometric function. When $n_2$ becomes large, the distribution of $T_0^2$ approaches that of $\chi^2$ based on $mn_1$ degrees of freedom. Ito [9] has derived asymptotic expansions both for the cumulative distribution function (cdf) of $T_0^2,$ and for the percentiles of $T_0^2$ in terms of the corresponding $\chi^2_{mn_1}$ percentiles. Other approximations to the distribution requiring large $n_2$ for validity have been obtained by Pillai and Samson [12]. These authors have used the method of fitting a Pearson curve by means of moment quotients to tabulate upper 5% and 1% points for $m = 2, 3, 4$. The exact distribution of $T$ over the range $0 \leqq T < 1$ has been obtained in the general non-central case by Constantine [3], using the methods of zonal polynomials and hypergeometric functions of matrix argument developed by James and Constantine ([2] and [10], for example). Constantine's solution has the form \begin{equation*}\tag{1.3}f(T) = \lbrack\Gamma_m(\frac{1}{2}(n_1 + n_2))/\Gamma(\frac{1}{2}mn_1)\Gamma_m(\frac{1}{2}n_2)\rbrack T ^{\frac{1}{2}mn_1-1}\mathscr{P}(T),\end{equation*} where $\mathscr{P}(T)$ is a power series in $T$ convergent in the unit circle, and \begin{equation*}\tag{1.4}\Gamma_m(z) = \pi^{\frac{1}{4}m(m-1)} \prod^{m-1}_{i=0} \Gamma(z - \frac{1}{2}i).\end{equation*} In Section 2 of the present paper, it is shown that in the null case the density function $f(T)$ (or rather, its analytic continuation into the complex $T$-plane) satisfies an ordinary linear differential equation of degree $m$ of Fuchsian type, having regular singularities at $T = 0, -1, \cdots, -m$ and infinity. More specifically, an equivalent first-order system is obtained, and the problem is most conveniently treated in this form. Constantine's series (1.3) in the null case is shown in Section 3 to be the relevant solution for $f(T)$ in the neighborhood of the regular singularity at $T = 0$. The differential equations lead to convenient recurrence relations for the coefficients in $\mathscr{P}(T)$. In Section 4 an alternative derivation of Ito's asymptotic formula is presented. Preliminary results are then given (Section 5) for the regular singularity at $T = \infty$, and a heuristic treatment of the limiting distribution as $n_1 \rightarrow \infty$ is presented in Section 6. Finally, it is shown in Section 7 that the moments of $T$ may be obtained from the differential equations for the Laplace transform of $f(T)$ given in Section 1. One objective in deriving the differential equations for $f(T)$ has been to obtain a convenient exact method for computing the distribution and its percentiles. This work is in progress, and it is hoped that results will be available shortly.