Matric variate Pearson type II-Riesz distribution

1 Introduction

Matrix distribution theory has transformed the vision of statistics applications in the last century; the usual real and univariate setting was generalised for large random objects in the standard numerical fields, constituting powerful techniques used in several branches of knowledge. That tendency allowed that any imaginable approach and application of univariate statistics could be taught in a greater framework. As usual, matrix generalisations based on real Gaussian models appeared in numerous papers over the past 50 years; verbatim copies of those classical results were translated separately into the complex and quaternion cases, without showing the underlying fact explained by certain abstract theories of mathematics.

Extensions to matrix variate non-Gaussian models opened an interesting perspective in the context of generalised invariant statistics and propitiate some strong results which are now widely applied in recent areas such as statistical shape theory and MANOVA. For example, transition to unified studies, of special families of distributions such as Pearson type II, took several years and required strong mathematical theories, which were usually out of the scope of statistical papers. In this case, the addressed distribution emerges in the following context: let $\mathbf{X}$ and ${\mathbf{U}}_1$ be random matrices independently distributed as matrix multivariate normal distribution and a Wishart distribution, respectively; then the random matrix $\mathbf{R}={\mathbf{L}}^{-1}\mathbf{X}$ , where $\mathbf{L}$ is any square root of $\mathbf{U}={\mathbf{L}}^{\ast }\mathbf{L}={\mathbf{U}}_1+{\mathbf{X}}^{\ast }\mathbf{X}$ , has a matric variate Pearson type II distribution. In the real case under normality, the matric variate Pearson type II distribution (also known as matric variate inverted T distribution) was studied separately by Khatri (1959), Dickey (1967) and Press (1982). Recently, in a general and unified setting, Díaz-García and Gutiérrez-Jáimez (2012) studied the real, complex, quaternion and octonion versions of this distribution.

Within the context of Bayesian inference, the posterior mean and generalised maximum likelihood estimators were found by Fang and Li (1999), assuming a matric variate Pearson type II distribution as the sampling model, and considering the posterior and marginal laws as the corresponding noninformative prior distributions. Meanwhile, with the frequentist approach, Díaz-García and Gutiérrez-Jáimez (2006) and Kotz and Nadarajah (2004) studied the normal regression based on Studentised errors.

In multivariate analysis the matric variate Pearson type II distribution is a source of interesting potential studies, for example, let $\mathbf{R}$ be a matric variate Pearson type II random matrix, then ${\mathbf{R}}^{\ast }\mathbf{R}$ follows a matrix multivariate beta type I distribution, a law which plays a fundamental role in MANOVA theory, see Khatri (1959, 1970) and Muirhead (1982).

A family of distributions on symmetric cones, termed the matrix multivariate Riesz distributions, was first introduced by Hassairi and Lajmi (2001) under the name of Riesz natural exponential family (Riesz NEF); it was based on a special case of the so-termed Riesz measure from Faraut and Korányi (1994, p. 137), going back to Riesz (1949). This Riesz distribution generalises the matrix multivariate gamma and Wishart distributions, containing them as particular cases. Subsequently, Díaz-García (2015c,a) proposed two versions of the Riesz distribution and two generalisations of a class of Kotz type distributions. The addressed general laws are termed matrix multivariate Kotz–Riesz distribution and contains the matrix multivariate normal distribution as a particular case.

With a similar philosophy, we can search a generalisation of the matric variate Pearson type II distribution, in the following way: let $\mathbf{R}={\mathbf{XL}}^{-1}$ , where $\mathbf{L}$ is a upper triangular matrix such that $\mathbf{U}={\mathbf{L}}^{\ast }\mathbf{L}={\mathbf{U}}_1+{\mathbf{X}}^{\ast }\mathbf{X}$ ; if we assume that $\mathbf{X}$ and ${\mathbf{U}}_1$ are independently distributed matrix multivariate Kotz–Riesz distribution and matrix multivariate Riesz distribution, then we can derive the required distribution of $\mathbf{R}$ , which will be called the matric variate Pearson type II-Riesz distribution.

In the last 30 years, the theory of random matrix distributions has reached a substantial development involving certain special areas of mathematics. Essentially, these advances have been archived through two approaches based on the theory of Jordan algebras and the theory of real normed division algebras. A basic source of the mathematical tools of theory of random matrices distributions under Jordan algebras can be found in Faraut and Korányi (1994); and specifically, some works in the context of theory of random matrix distributions based on Jordan algebras are provided in Massam (1994), Casalis and Letac (1996), Hassairi and Lajmi (2001) and Hassairi et al. (2005), and the references therein. Parallel results on theory of random matrix distributions based on real normed division algebras have been also developed in random matrix theory and statistics, see Gross and Richards (1987), Dumitriu (2002), Forrester (2005) and Díaz-García and Gutiérrez-Jáimez (2011, 2013), among others. Instead of using Jordan algebras, Ishi (2000) and Boutouria and Hassiri (2009) studied several basic properties of the matrix multivariate Riesz distribution under the structure theory of normal j-algebras and theory of Vinberg algebras, respectively.

Finally, the application of some particular fields as the octonions seems to be unclear at present. An excellent review of the history, construction and properties of octonions can be found in Baez (2002); moreover, that author comments:

“Their relevance to geometry was quite obscure until 1925, when Élie Cartan described ‘triality’ – the symmetry between vector and spinors in 8-dimensional Euclidian space. Their potential relevance to physics was noticed in a 1934 paper by Jordan, von Neumann and Wigner on the foundations of quantum mechanics…Work along these lines continued quite slowly until the 1980s, when it was realised that the octionions explain some curious features of string theory… However, there is still no proof that the octonions are useful for understanding the real world. We can only hope that eventually this question will be settled one way or another.”

For the sake of completeness, the octonions will be considered in this work, but we must recognise that the application of the associated results can only be conjectured. Even so, some expectations are emerging, for example, Forrester (2005, Section 1.4.5, pp. 22-24) proved that the bi-dimensional eigenvalue density function of a $2\times 2$ octonionic matrix Gaussian ensemble is obtained from the eigenvalue general joint density function of a Gaussian ensemble with $m=2$ and $\beta =8$ , see notation in Section 2. Moreover, according to Faraut and Korányi (1994) and Sawyer (1997), it is easy to check that the results of this work are valid for the algebra of Albert, i.e., when the involved hermitian matrices or certain products of hermitian matrices are $3\times 3$ octonionic matrices.

The present paper is organised as follows: basic concepts and notations of abstract algebra and Jacobians are summarised in Section 2; and, definitions and properties of the nonsingular central matric variate Pearson type II-Riesz and beta type I distributions are studied in Section 3. We emphasise that the results are derived in the context of real normed division algebras, a useful integrated and unified approach recently implemented in matrix distribution theory.

2 Preliminary results

A detailed discussion of real normed division algebras can be found in Baez (2002) and Neukirch et al. (1990). For convenience, we shall introduce some notation, although in general we adhere to standard notation forms.

Let $\mathbb{F}$ be a field. An algebra $\mathfrak{A}$ over $\mathbb{F}$ is a pair $(\mathfrak{A}\text{;}m)$ , where $\mathfrak{A}$ is a finite-dimensional vector space over $\mathbb{F}$ and multiplication $m:\mathfrak{A}\times \mathfrak{A}\to A$ is an $\mathbb{F}$ -bilinear map; that is, for all $\lambda \in \mathbb{F}\text{,}x\text{,}y\text{,}z\in \mathfrak{A}$ , $$\begin{array}{cc}\hfill & m(x\text{,}\lambda y+z)=\lambda m(x\text{;}y)+m(x\text{;}z)\text{,}\hfill \\ \hfill & m(\lambda x+y\text{;}z)=\lambda m(x\text{;}z)+m(y\text{;}z)\text{.}\hfill \end{array}$$ Two algebras $(\mathfrak{A}\text{;}m)$ and $(\mathfrak{E}\text{;}n)$ over $\mathbb{F}$ are said to be isomorphic if there is an invertible map $\varphi :\mathfrak{A}\to \mathfrak{E}$ such that for all $x\text{,}y\in \mathfrak{A}$ , $$\varphi (m(x\text{,}y))=n(\varphi (x)\text{,}\varphi (y))\text{.}$$ By simplicity, we write $m(x\text{;}y)=\mathit{xy}$ for all $x\text{,}y\in \mathfrak{A}$ .

Let $\mathfrak{A}$ be an algebra over $\mathbb{F}$ . Then $\mathfrak{A}$ is said to be

1. alternative if $x(\mathit{xy})=(\mathit{xx})y$ and $x(\mathit{yy})=(\mathit{xy})y$ for all $x\text{,}y\in \mathfrak{A}$ ,

2. associative if $x(\mathit{yz})=(\mathit{xy})z$ for all $x\text{,}y\text{,}z\in \mathfrak{A}$ ,

3. commutative if $\mathit{xy}=\mathit{yx}$ for all $x\text{,}y\in \mathfrak{A}$ , and

4. unital if there is a $1\in \mathfrak{A}$ such that $x1=x=1x$ for all $x\in \mathfrak{A}$ .

An algebra $\mathfrak{A}$ over $\mathbb{F}$ is said to be a division algebra if $\mathfrak{A}$ is nonzero and $\mathit{xy}=0_{\mathfrak{A}}\Rightarrow x=0_{\mathfrak{A}}$ or $y=0_{\mathfrak{A}}$ for all $x\text{,}y\in \mathfrak{A}$ .

The term “division algebra”, comes from the following proposition, which shows that, in such an algebra, left and right division can be unambiguously performed.

Let $\mathfrak{A}$ be an algebra over $\mathbb{F}$ . Then $\mathfrak{A}$ is a division algebra if, and only if, $\mathfrak{A}$ is nonzero and for all $a\text{,}b\in \mathfrak{A}$ , with $b\ne 0_{\mathfrak{A}}$ , the equations $\mathit{bx}=a$ and $\mathit{yb}=a$ have unique solutions $x\text{,}y\in \mathfrak{A}$ .

In the sequel we assume $\mathbb{F}=\mathfrak{R}$ and consider classes of division algebras over $\mathfrak{R}$ or “real division algebras” for short.

We introduce the algebras of real numbers $\mathfrak{R}$ , complex numbers $\mathfrak{C}$ , quaternions $\mathfrak{H}$ and octonions $\mathfrak{O}$ . Then, if $\mathfrak{A}$ is an alternative real division algebra, then $\mathfrak{A}$ is isomorphic to $\mathfrak{R}$ , $\mathfrak{C}\text{,}\mathfrak{H}$ or $\mathfrak{O}$ .

Let $\mathfrak{A}$ be a real division algebra with identity 1. Then $\mathfrak{A}$ is said to be normed if there is an inner product $(·\text{,}·)$ on $\mathfrak{A}$ such that $$(\mathit{xy}\text{,}\mathit{xy})=(x\text{,}x)(y\text{,}y)\mathrm{forall}x\text{,}y\in \mathfrak{A}\text{.}$$

Let $\mathfrak{A}$ be a division algebra over the real numbers. Then $\mathfrak{A}$ has dimension either 1, 2, 4 or 8. In other branches of mathematics, the parameters $\alpha =2/\beta$ and $t=\beta /4$ are used, see Edelman and Rao (2005) and Khatri (1984), respectively.

Finally, observe that

• $\mathfrak{R}$ is a real commutative associative normed division algebra,

• $\mathfrak{C}$ is a commutative associative normed division algebra,

• $\mathfrak{H}$ is an associative normed division algebra,

• $\mathfrak{O}$ is an alternative normed division algebra.

Let ${\mathcal{L}}_{n\text{,}m}^{\beta }$ be the set of all $n\times m$ matrices of rank $m⩽n$ over $\mathfrak{A}$ with m distinct positive singular values, where $\mathfrak{A}$ denotes a real finite-dimensional normed division algebra. Let ${\mathfrak{A}}^{n\times m}$ be the set of all $n\times m$ matrices over $\mathfrak{A}$ . The dimension of ${\mathfrak{A}}^{n\times m}$ over $\mathfrak{R}$ is $\beta \mathit{mn}$ . Let $\mathbf{A}\in {\mathfrak{A}}^{n\times m}$ , then ${\mathbf{A}}^{\ast }={\overline{\mathbf{A}}}^T$ denotes the usual conjugate transpose.

Table 1 sets out the equivalence among the same concepts in the four normed division algebras.

We denote by ${\mathfrak{S}}_m^{\beta }$ the real vector space of all $\mathbf{S}\in {\mathfrak{A}}^{m\times m}$ such that $\mathbf{S}={\mathbf{S}}^{\ast }$ . In addition, let ${\mathfrak{P}}_m^{\beta }$ be the cone of positive definite matrices $\mathbf{S}\in {\mathfrak{A}}^{m\times m}$ . Thus, ${\mathfrak{P}}_m^{\beta }$ consist of all matrices $\mathbf{S}={\mathbf{X}}^{\ast }\mathbf{X}$ , with $\mathbf{X}\in {\mathfrak{L}}_{n\text{,}m}^{\beta }$ ; then ${\mathfrak{P}}_m^{\beta }$ is an open subset of ${\mathfrak{S}}_m^{\beta }$ .

Let ${\mathfrak{D}}_m^{\beta }$ consisting of all $\mathbf{D}\in {\mathfrak{A}}^{m\times m}$ , $\mathbf{D}=\mathrm{diag}(d_1\text{,}\dots \text{,}{d}_m)$ and let ${\mathfrak{T}}_U^{\beta }(m)$ be the subgroup of all upper triangular matrices $\mathbf{T}\in {\mathfrak{A}}^{m\times m}$ such that $t_{\mathit{ij}}=0$ for $1<i<j⩽m$ .

For any matrix $\mathbf{X}\in {\mathfrak{A}}^{n\times m}\text{,}d\mathbf{X}$ denotes thematrix of differentials $({\mathit{dx}}_{\mathit{ij}})$ . Finally, we define the measure or volume element $(d\mathbf{X})$ when $\mathbf{X}\in {\mathfrak{A}}^{n\times m}\text{,}{\mathfrak{S}}_m^{\beta }$ , ${\mathfrak{D}}_m^{\beta }$ or ${\mathcal{V}}_{m\text{,}n}^{\beta }$ , see Díaz-García and Gutiérrez-Jáimez (2011, 2013).

If $\mathbf{X}\in {\mathfrak{A}}^{n\times m}$ then $(d\mathbf{X})$ (the Lebesgue measure in ${\mathfrak{A}}^{n\times m}$ ) denotes the exterior product of the $\beta \mathit{mn}$ functionally independent variables $$(d\mathbf{X})={\displaystyle \underset{i=1}{\overset{n}{\bigwedge }}}{\displaystyle \underset{j=1}{\overset{m}{\bigwedge }}}{\mathit{dx}}_{\mathit{ij}}\mathrm{where}{\mathit{dx}}_{\mathit{ij}}={\displaystyle \underset{k=1}{\overset{\beta }{\bigwedge }}}{\mathit{dx}}_{\mathit{ij}}^{(k)}\text{.}$$

If $\mathbf{S}\in {\mathfrak{S}}_m^{\beta }$ (or $\mathbf{S}\in {\mathfrak{T}}_U^{\beta }(m)$ with $t_{\mathit{ii}}>0\text{,}i=1\text{,}\dots \text{,}m$ ) then $(d\mathbf{S})$ (the Lebesgue measure in ${\mathfrak{S}}_m^{\beta }$ or in ${\mathfrak{T}}_U^{\beta }(m)$ ) denotes the exterior product of the $m(m-1)\beta /2+m$ functionally independent variables, $$(d\mathbf{S})={\displaystyle \underset{i=1}{\overset{m}{\bigwedge }}}{\mathit{ds}}_{\mathit{ii}}{\displaystyle \underset{i>j}{\overset{m}{\bigwedge }}}{\displaystyle \underset{k=1}{\overset{\beta }{\bigwedge }}}{\mathit{ds}}_{\mathit{ij}}^{(k)}\text{.}$$ Observe that the Lebesgue measure $(d\mathbf{S})$ requires that $\mathbf{S}\in {\mathfrak{P}}_m^{\beta }$ , i.e., $\mathbf{S}$ must be a non singular Hermitian matrix (Hermitian definite positive matrix).

If $\mathbf{\Lambda }\in {\mathfrak{D}}_m^{\beta }$ then $(d\mathbf{\Lambda })$ (the Lebesgue measure in ${\mathfrak{D}}_m^{\beta }$ ) denotes the exterior product of the $\beta m$ functionally independent variables $$(d\mathbf{\Lambda })={\displaystyle \underset{i=1}{\overset{n}{\bigwedge }}}{\displaystyle \underset{k=1}{\overset{\beta }{\bigwedge }}}d{\lambda }_i^{(k)}\text{.}$$ If ${\mathbf{H}}_1\in {\mathcal{V}}_{m\text{,}n}^{\beta }$ then $$({\mathbf{H}}_1^{\ast }d{\mathbf{H}}_1)={\displaystyle \underset{i=1}{\overset{m}{\bigwedge }}}{\displaystyle \underset{j=i+1}{\overset{n}{\bigwedge }}}{\mathbf{h}}_j^{\ast }d{\mathbf{h}}_i\text{.}$$ where $\mathbf{H}={({\mathbf{H}}_1^{\ast }|{\mathbf{H}}_2^{\ast })}^{\ast }={({\mathbf{h}}_1\text{,}\dots \text{,}{\mathbf{h}}_m|{\mathbf{h}}_{m+1}\text{,}\dots \text{,}{\mathbf{h}}_n)}^{\ast }\in {\mathfrak{U}}^{\beta }(n)$ . It can be proved that this differential form does not depend on the choice of the ${\mathbf{H}}_2$ matrix. When $n=1\text{;}{\mathcal{V}}_{m\text{,}1}^{\beta }$ defines the unit sphere in ${\mathfrak{A}}^m$ , an $(m-1)\beta$ -dimensional surface in ${\mathfrak{A}}^m$ . When $n=m$ and denoting ${\mathbf{H}}_1$ by $\mathbf{H}$ , $(\mathbf{H}d{\mathbf{H}}^{\ast })$ is termed the Haar measure on ${\mathfrak{U}}^{\beta }(m)$ .

The surface area or volume of the Stiefel manifold ${\mathcal{V}}_{m\text{,}n}^{\beta }$ is

In other branches of mathematics the highest weight vector $q_{\kappa }(\mathbf{A})$ is also termed the generalised power of $\mathbf{A}$ and is denoted as ${\mathrm{\Delta }}_{\kappa }(\mathbf{A})$ , see Faraut and Korányi (1994) and Hassairi and Lajmi (2001).

Several properties of $q_{\kappa }(\mathbf{A})$ can be easily obtained, a list of them is given next:

1. Let $\mathbf{A}={\mathbf{L}}^{\ast }\mathbf{DL}$ be the L’DL decomposition of $\mathbf{A}\in {\mathfrak{P}}_m^{\beta }$ , where $\mathbf{L}\in {\mathfrak{T}}_U^{\beta }(m)$ with $l_{\mathit{ii}}=1\text{,}i=1\text{,}2\text{,}\dots \text{,}m$ and $\mathbf{D}=\mathrm{diag}({\lambda }_1\text{,}\dots \text{,}{\lambda }_m)\text{,}{\lambda }_i⩾0\text{,}i=1\text{,}2\text{,}\dots \text{,}m$ . Then

   (5)
   $$q_{\kappa }(\mathbf{A})={\displaystyle \prod _{i=1}^m}{\lambda }_i^{k_i}\text{.}$$

2. (6)
   $$q_{\kappa }({\mathbf{A}}^{-1})=q_{-{\kappa }^{\ast }}^{\ast }(\mathbf{A})\text{,}$$ where ${\kappa }^{\ast }=(k_m\text{,}{k}_{m-1}\text{,}\dots \text{,}{k}_1)\text{,}-{\kappa }^{\ast }=(-k_m\text{,}-k_{m-1}\text{,}\dots \text{,}-k_1)$ ,
   (7)
   $$q_{\kappa }^{\ast }(\mathbf{A})=|{\mathbf{A}}_m{|}^{k_m}{\displaystyle \prod _{i=1}^{m-1}}|{\mathbf{A}}_i{|}^{k_i-k_{i+1}}$$ and
   (8)
   $$q_{\kappa }^{\ast }(\mathbf{A})={\displaystyle \prod _{i=1}^m}{\lambda }_i^{k_{m-i+1}}\text{,}$$ see Faraut and Korányi (1994, pp. 126–127 and Proposition VII.1.5).

   Alternatively, let $\mathbf{A}={\mathbf{T}}^{\ast }\mathbf{T}$ the Cholesky decomposition of matrix $\mathbf{A}\in {\mathfrak{P}}_m^{\beta }$ , with $\mathbf{T}=(t_{\mathit{ij}})\in {\mathfrak{T}}_U^{\beta }(m)$ , then ${\lambda }_i=t_{\mathit{ii}}^2\text{,}{t}_{\mathit{ii}}⩾0\text{,}i=1\text{,}2\text{,}\dots \text{,}m$ . See Hassairi and Lajmi (2001, p. 931, first paragraph), Hassairi et al. (2005, p. 390, lines -11 to -16) and Kołodziejek (2014, p.5, lines 1-6).

3. if $\kappa =(p\text{,}\dots \text{,}p)$ , then

   (9)
   $$q_{\kappa }(\mathbf{A})=|\mathbf{A}{|}^p\text{,}$$ in particular if $p=0$ , then $q_{\kappa }(\mathbf{A})=1$ .

4. if $\tau =(t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_m)\text{,}{t}_1⩾t_2⩾\cdots ⩾t_m⩾0$ , then

   (10)
   $$q_{\kappa +\tau }(\mathbf{A})=q_{\kappa }(\mathbf{A})q_{\tau }(\mathbf{A})\text{,}$$ in particular if $\tau =(p\text{,}p\text{,}\dots \text{,}p)$ , then
   (11)
   $$q_{\kappa +\tau }(\mathbf{A})\equiv q_{\kappa +p}(\mathbf{A})=|\mathbf{A}{|}^p{q}_{\kappa }(\mathbf{A})\text{.}$$

5. Finally, for $\mathbf{B}\in {\mathfrak{T}}_U^{\beta }(m)$ in such a manner that $\mathbf{C}={\mathbf{B}}^{\ast }\mathbf{B}\in {\mathfrak{S}}_m^{\beta }$ ,

   (12)
   $$q_{\kappa }({\mathbf{B}}^{\ast }\mathbf{AB})=q_{\kappa }(\mathbf{C})q_{\kappa }(\mathbf{A})$$ and
   (13)
   $$q_{\kappa }({\mathbf{B}}^{\ast -1}{\mathbf{AB}}^{-1})={(q_{\kappa }(\mathbf{C}))}^{-1}{q}_{\kappa }(\mathbf{A})=q_{-\kappa }(\mathbf{C})q_{\kappa }(\mathbf{A})\text{,}$$ see Hassairi et al. (2008, p. 776, Eq. (2.1)).

In (3), ${[a]}_{\kappa }^{\beta }$ denotes the generalised Pochhammer symbol of weight $\kappa$ , defined as $$\begin{array}{cc}\hfill & {[a]}_{\kappa }^{\beta }={\displaystyle \prod _{i=1}^m}{(a-(i-1)\beta /2)}_{k_i}\hfill \\ \hfill & =\frac{{\pi }^{m(m-1)\beta /4}{\prod }_{i=1}^m\mathrm{\Gamma }[a+k_i-(i-1)\beta /2]}{{\mathrm{\Gamma }}_m^{\beta }[a]}\hfill \\ \hfill & =\frac{{\mathrm{\Gamma }}_m^{\beta }[a\text{,}\kappa ]}{{\mathrm{\Gamma }}_m^{\beta }[a]}\text{,}\hfill \end{array}$$ where $\mathrm{Re}(a)>(m-1)\beta /2-k_m$ and $${(a)}_i=a(a+1)\cdots (a+i-1)$$ is the standard Pochhammer symbol.

An alternative definition of the generalised gamma function of weight $\kappa$ is proposed by Khatri, 1966:

Consider also the following generalised beta functions termed, generalised c-beta function, see Faraut and Korányi (1994, p. 130) and Díaz-García (2015b), $$\begin{array}{cc}\hfill & {\mathcal{B}}_m^{\beta }[a\text{,}\kappa \text{;}b\text{,}\tau ]\hfill \\ \hfill & ={\int }_{\mathbf{0}<\mathbf{S}<{\mathbf{I}}_m}|\mathbf{S}{|}^{a-(m-1)\beta /2-1}{q}_{\kappa }(\mathbf{S})|{\mathbf{I}}_m-\mathbf{S}{|}^{b-(m-1)\beta /2-1}{q}_{\tau }({\mathbf{I}}_m-\mathbf{S})(d\mathbf{S})\hfill \\ \hfill & ={\int }_{\mathbf{R}\in {\mathfrak{P}}_m^{\beta }}|\mathbf{R}{|}^{a-(m-1)\beta /2-1}{q}_{\kappa }(\mathbf{R})|{\mathbf{I}}_m+\mathbf{R}{|}^{-(a+b)}{q}_{-(\kappa +\tau )}({\mathbf{I}}_m+\mathbf{R})(d\mathbf{R})\hfill \\ \hfill & =\frac{{\mathrm{\Gamma }}_m^{\beta }[a\text{,}\kappa ]{\mathrm{\Gamma }}_m^{\beta }[b\text{,}\tau ]}{{\mathrm{\Gamma }}_m^{\beta }[a+b\text{,}\kappa +\tau ]}\text{,}\hfill \end{array}$$ where $\kappa =(k_1\text{,}{k}_2\text{,}\dots \text{,}{k}_m)\in {\mathfrak{R}}^m\text{,}\tau =(t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_m)\in {\mathfrak{R}}^m$ , Re $(a)>(m-1)\beta /2-k_m$ and Re $(b)>(m-1)\beta /2-t_m$ . Similarly defined is the generalised k-beta function as, see Díaz-García (2015b), $$\begin{array}{cc}\hfill & {\mathcal{B}}_m^{\beta }[a\text{,}-\kappa \text{;}b\text{,}-\tau ]\hfill \\ \hfill & ={\int }_{\mathbf{0}<\mathbf{S}<{\mathbf{I}}_m}|\mathbf{S}{|}^{a-(m-1)\beta /2-1}{q}_{\kappa }({\mathbf{S}}^{-1})|{\mathbf{I}}_m-\mathbf{S}{|}^{b-(m-1)\beta /2-1}{q}_{\tau }\left({({\mathbf{I}}_m-\mathbf{S})}^{-1}\right)(d\mathbf{S})\hfill \\ \hfill & ={\int }_{\mathbf{R}\in {\mathfrak{P}}_m^{\beta }}|\mathbf{R}{|}^{a-(m-1)\beta /2-1}{q}_{\kappa }({\mathbf{R}}^{-1})|{\mathbf{I}}_m+\mathbf{R}{|}^{-(a+b)}{q}_{-(\kappa +\tau )}\left({({\mathbf{I}}_m+\mathbf{R})}^{-1}\right)(d\mathbf{R})\hfill \\ \hfill & =\frac{{\mathrm{\Gamma }}_m^{\beta }[a\text{,}-\kappa ]{\mathrm{\Gamma }}_m^{\beta }[b\text{,}-\tau ]}{{\mathrm{\Gamma }}_m^{\beta }[a+b\text{,}-\kappa -\tau ]}\text{,}\hfill \end{array}$$ where $\kappa =(k_1\text{,}{k}_2\text{,}\dots \text{,}{k}_m)\in {\mathfrak{R}}^m\text{,}\tau =(t_1\text{,}{t}_2\text{,}\dots \text{,}{t}_m)\in {\mathfrak{R}}^m$ , Re $(a)>(m-1)\beta /2+k_1$ and Re $(b)>(m-1)\beta /2+t_1$ .

Finally, the following Jacobians involving the $\beta$ parameter, reflects the generalised power of the algebraic technique; they can be seen as extensions of the full derived and unconnected results in the real, complex or quaternion cases, see Faraut and Korányi (1994) and Díaz-García and Gutiérrez-Jáimez (2011). These results are the base for several matrix and matric variate generalised analyses.

3 Matric variate Pearson type II-Riesz distribution

Two versions of the matric variate Pearson type II-Riesz distributions and the corresponding generalised beta type I distributions are obtained in this section.

A discussion of Riesz distribution may be found in Hassairi and Lajmi (2001) and Díaz-García (2015a); and a description of Kotz–Riesz distribution is given in Díaz-García (2015b). For convenience, we adhere to standard notation stated in Díaz-García (2015a,b). Now, consider the following two definitions.

An alternative way to define the matric variate Pearson type II-Riesz distributions is collected in the following result.