Multimatricvariate and multimatrix variate distributions based on elliptically contoured laws under real normed division algebras

A postmodern world of increasingly heuristic and interdisciplinary problems requires integrative solutions that do not always have the answer in the current frameworks of mathematics and statistics. Chaos theory as a paradigmatic element of phenomenal interrelation links more and more random variables. They are not only univariate and vector variables, but matrix variates, due to their great versatility in the description of variability and intrinsic multiple correlation. Likewise, the domain of the variables began to move from the real field to the remaining real normed division algebras, such as complex, quaternionic and octonionic. There the emerging models of multiple natural, exact and engineering sciences have their own scope and they are not restricted by the real field. Given multiple statistical models for marginal distributions in univariate and vector cases, the additional problem of parametric estimation arises from a set of samples provided by the scientific expert user of statistics. Then comes another challenge for statistics, which has historically been resolved by optimising likelihood functions as a joint law for sample distribution. The theory of copulas emerged as a possible solution to the problem of explaining a joint phenomenon of known interrelation. Countless link functions then appear for two variables, however, the intention of a dependent relationship is diluted when using the typical likelihood functions for estimating copula parameters from independent samples. The role of likelihood in the history of statistics is of such transcendence that it is a popular tool for estimating parameters; however, its profound simplicity based on sample independence is never debated. While copulas and similar theories represent the solution to the problem of dependence, the concept of likelihood function, defined as the product of the marginal densities, seems immutable and universal. Recently, the authors have begun the discussion of the role of classical likelihood functions in statistics, showing the differences in application in databases declared dependently probabilistic. Time series in particular are the source of the greatest discrepancy, given the historical effort in proposing models for variance and volatility. But maintaining the postulate of an estimate via likelihood functions on independent data is an obvious contradiction in the face of a temporal process that intrinsically is founded on dependency. A work that launched this new way of reframing likelihood functions appeared recently in Díaz-García et al. [7]. In a real data series, the discrepancy of likelihood function over independent time samples versus realistic likelihood function over dependent samples was observed. The study indicated that decisions diffused about the mean estimated from the database using independent likelihood were in the tails of the dependent expected likelihood distribution. The finding was possible thanks to the definition of the so-termed multivector variate distribution, a natural way of defining a likelihood function of dependent vector samples, further parameterised by a large class of elliptically contoured distributions that are isolated from the choices of popular models. Until then, the mutivector could be an approach to vector copulas, but the implemented theory is designed to address the matrix case, which is still an open problem for copula theory. Then, the matrix version appeared in the form of the so-termed classes of multimatricvariate distributions, which provided the likelihood function or joint matrix distributions with a family of elliptically contoured distributions with a variety of kurtosis and symmetries, see Díaz-García and Caro-Lopera [5]. The choice of distributions that are invariant under the class of elliptically contoured laws within the family of multimatricvariate and multimatrix variate distributions gives robustness to the analysis since it avoids the difficult paradigm of adjusting the link function, if we speak in parallel to the procedure. in vector copulas.

Multimatricvariate distributions specialise in distributions based on the determinant. Still to be defined a class of multiple distributions of random matrices that depended on the trace, the other function that has historically governed the theory of matrix distributions. Recently, the so-termeded multimatrix distributions appear in Díaz-García and Caro-Lopera [6]. Like the multimatricvariate distributions, they maintain the philosophy of probabilistic dependence, computability and a wide class of underlying distributions.

When we surpass the level of random vectors, where the most popular statistical techniques remain, such as copulas, we find immense difficulty for the calculation. We must observe the domain of the matrices, their transformations and Jacobians, and therefore their integration. It is generally difficult because it involves averaging over orthogonal groups, cones and hypercubes. The central, isotropic, non-central and non-isotropic cases also emerge. These integrations comprises the geometric filtering that comes for their definitions and factorisations.

At this point we find that the new multimatricvariate and multimatrix variate distributions on the real field satisfy the first 2 of the 3 conditions that we establish as a paradigm for a robust point of view of joint distributions, namely: 1) modeling of several dependent matrices capable of being combined with all univariate and/or vector variables governed by elliptically contoured models; 2) with strictly verifiable marginals under the trivial case of independence and testable from the dependent sample likelihood; and 3) founded on the same algebraic principles that allow them to be applied indistinctly from the real normed division algebras.

We now address the third desirable characteristic: its versatility in the four unique real normed division algebras.

Until a couple of decades ago it would be unthinkable to unify the theory of real random matrices with complex ones. History shows us that the theory of distributions of the central cases overflowed into an immense effort to characterize the indispensable elements of random matrices, namely: After its establishment, works on the complex case began to appear very slowly and completely conceptually distant from their real analogues. The so-termed statistical theory under real normed division algebras is to statistics what the expected theory of unification of forces is to physics. Its simplification is surprising in that all its results are reduced to modifying a beta parameter that goes from 1 (real) to 2 (complex) to 4 (quaternionic) to 8 (octonionic), see Díaz-García [4] were the transitions to complex, quaternionic and octonion are reached by changing the support group from orthogonal to unitary, compact symplectic or exceptional type.

If the real multimatricvariate distributions [5] and multimatrix variate distributions [6] articles are parallel compared with the results of the present work, an apparent repetition shall appear by its simplicity; Its notation makes the beta parameter open one of the four universes by simply changing the value, although understanding the profound difference that underpins them requires an extensive and difficult literature that began in other areas far from statistics. The non commutativity for quaternions and non associative for octonions promotes a deep research for some applications in those algebras.

We place the above discussion into the setting of two main problems that we shall address in this article.

The interest in multimatricvariate and multimatrix variate distributions has been motivated by the following two situations:

1. 1.

   In different areas of knowledge (such as Finance and Hydrology, among others), people are interested in simultaneously modeling two random variables, say $X$ and $Y$, which are suspected of not being probabilistically independent. On the one hand, the marginal distributions of each variable are known, whether they are $f_{X}(x)$ and $g_{Y}(y)$. Typically this problem has been approached assuming that the random variables $X$ and $Y$ are independent and, as a function of joint density of the two-dimensional vector $(X,Y)^{\prime}$, the product of the marginals, $r_{X,Y}(x,y)=f_{X}(x)g_{Y}(y)$, has been considered. Thus, for example, in this case the likelihood function, given the two-dimensional sample $(x_{1},y_{1}),\cdots,(x_{k},y_{k})$, denoted as $L(\boldmath{\theta};(x_{1},y_{1}),\cdots,(x_{k},y_{k}))$ is defined as

   	$\displaystyle L(\mathbf{\theta};(x_{1},y_{1}),\cdots,(x_{n},y_{n}))$	$\displaystyle=$	$\displaystyle\prod_{j=1}^{k}r_{X_{j},Y_{j}}(x_{j},y_{j})$
   		$\displaystyle=$	$\displaystyle\prod_{j=1}^{k}f_{X_{j}}(x_{j})g_{Y_{j}}(y_{j})$

   For some parameter vector $\boldmath{\theta}\in\Re^{p}$ which is part of the density $r_{X,Y}(x,y)$.

   Alternatively, based on variable changes on a set of independent random variables, the random vector $(X,Y)^{\prime}$ was generated, where now the variables $X$ and $Y$ are not independent and their density function is known joint $t_{X,Y}(x,y)\neq f_{X}(x)g_{Y}(y)$, such that

   $$f_{X}(x)=\int_{\mathfrak{\Re}}t_{X,Y}(x,y)(dy){\mbox{\hskip 14.22636pt and \hskip 14.22636pt}}g_{Y}(y)=\int_{\Re}t_{X,Y}(x,y)(dx).$$

   Under this approach we have that

   $$L(\boldmath{\theta};(x_{1},y_{1}),\cdots,(x_{k},y_{k}))=\prod_{j=1}^{k}t_{X_{j},Y_{j}}(x_{j},y_{j})$$

   See Libby and Novick [24], Chen and Novick [3], Olkin and Liu [28], Nadarajah [26, 27] and Sarabia et al. [30] among many others.

   This situation also occurs in other multivariate problems. Then, parallel solutions were proposed in the vector and matrix cases, giving foothold to the study of bi-matrix variate distributions in the real and complex cases. In the last case joint distributions of random matrices, say $\mathbf{X}$ and $\mathbf{Y}$ dependent with joint density function, termed bimatrix variate distribution, are proposed such that the marginal densities of $\mathbf{X}$ and $\mathbf{Y}$ are the usual assumptions, see Olkin and Rubin [29], Díaz-García, and Gutiérrez-Jáimez [8, 9, 11], Bekker et al. [1], and [16] and references therein.

2. 2.

   In another case, we are interested in defining the likelihood function by a joint function of the sample, but which is not defined as the product of the marginals, that is, the sample is not independent. In the univariate problem, one answer considers the elliptically contoured distribution of the vector $(X_{1},\dots,X_{k})^{\prime}$ as a likelihood function, noting that in reality the elliptically contoured distribution actually defines a distribution family, see Fang et al. [18], Fang and Zhang [17], Gupta and Varga [21] and the reference therein.

   Based on the family of matrix variate elliptically contoured distributions, the multimatrix variate and multimatricvariate distributions were proposed as a generalisation of the bi-matrix variate distributions, which are defined as the joint distribution of the dependent random matrices $\mathbf{X}_{1},\dots,\mathbf{X}_{k}$, see Díaz-García et al. [7], and Díaz-García and Caro-Lopera [5, 6]. Thus, the multimatrix variate and multimatricvariate distributions can be used as likelihood functions for a sample of dependent random matrices with certain (usual) marginal distributions. Thus, the likelihood function of the sample $\mathbf{X}_{1},\dots,\mathbf{X}_{k}$ is defined as

   $$L(\mathbf{\Theta};\mathbf{X}_{1},\dots,\mathbf{X}_{k})=f_{\mathbf{X}_{1},\dots,\mathbf{X}_{k}}(\mathbf{X}_{1},\dots,\mathbf{X}_{k}).$$

Under the theory of multimatrix matrix or multimatricvariate distributions, each matrix $\mathbf{X}_{j}$, $j=1,\dots,k$ into the density $f_{\mathbf{X}_{1},\dots,\mathbf{X}_{k}}(\mathbf{X}_{1},\dots,\mathbf{X}_{k})$ can follow a different marginal distribution. This answers the following problem under an independent or dependent samples: Suppose that we have a matrix random sample as follows:

And assume that the matrix $\mathbf{\Theta}$ contains the parameters of interest. Then the likelihood function $L(\mathbf{\Theta};\cdot)$ can be defined as:

In the bi-matrix variate case, these problems have been studied in the real and complex cases, each giving rise to a series of non correlated publications. Fortunately, in terms of the theory of the real normed division algebras, a unification of the real and complex cases is possible. And an extension to the quaternionic and octonionic algebras is also feasible. It is worth mentioning that the octornionic case is still under research. At this time, they are valid for $2\times 2$ octornionic matrices and in general it can only be conjectured that they may be valid.

In the present work, the multimatrix variate and multimatricvariate distributions are studied for matrix arguments which elements belong to the real normed division algebras. A brief description of the notation and some Jacobians for real normed division algebras is presented in Section 2. In addition, two more Jacobians are obtained and the definition of the matrix variate elliptically contoured distribution for real normed division algebras is presented. The main results on multimatrix variate and multimatricvariate distributions for real normed division algebras are obtained in Section 3. Some properties and extensions of multimatrix variate and multimatricvariate distribution with more than two different types of distributions in their arguments are studied in Section 4. An example in the quaternionic case is full derived in Section 5.

A detailed discussion of real normed division algebras may be found in Baez [2]. For convenience, we shall introduce some notations, although in general we adhere to standard notations.

A vector space is always a finite-dimensional module over the field of real numbers. An algebra $\mathfrak{F}$ is a vector space that is equipped with a bilinear map $m:\mathfrak{F}\times\mathfrak{F}\rightarrow\mathfrak{F}$ termed multiplication and a nonzero element $1\in\mathfrak{F}$ termed the unit such that $m(1,a)=m(a,1)=1$. As usual, we abbreviate $m(a,b)=ab$ as $ab$. We do not assume $\mathfrak{F}$ associative. Given an algebra, we freely think of real numbers as elements of this algebra via the map $\omega\mapsto\omega 1$.

An algebra $\mathfrak{F}$ is a division algebra if given $a,b\in\mathfrak{F}$ with $ab=0$, then either $a=0$ or $b=0$. Equivalently, $\mathfrak{F}$ is a division algebra if the operation of left and right multiplications by any nonzero element is invertible. A normed division algebra is an algebra $\mathfrak{F}$ that is also a normed vector space with $||ab||=||a||||b||$. This implies that $\mathfrak{F}$ is a division algebra and that $||1||=1$.

There are exactly four normed division algebras: real numbers ($\Re$), complex numbers ($\mathfrak{C}$), quaternions ($\mathfrak{H}$) and octonions ($\mathfrak{O}$), see Baez [2]. Taking into account that $\Re$, $\mathfrak{C}$, $\mathfrak{H}$ and $\mathfrak{O}$ are the only normed division algebras; moreover, they are the only alternative division algebras, and all division algebras have a real dimension of $1,2,4$ or $8$, which is denoted by $\beta$, see Baez [2, Theorems 1, 2 and 3]. In other branches of mathematics, the parameter $\alpha=2/\beta$ is used, see Edelman and Rao [15].

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

The set of matrices $\mathbf{H}_{1}\in\mathfrak{F}^{n\times m}$ such that $\mathbf{H}_{1}^{H}\mathbf{H}_{1}=\mathbf{I}_{m}$ is a manifold denoted ${\mathcal{V}}_{m,n}^{\beta}$, is termed the Stiefel manifold ($\mathbf{H}_{1}$ is also known as semi-orthogonal ($\beta=1$), semi-unitary ($\beta=2$), semi-symplectic ($\beta=4$) and semi-exceptional type ($\beta=8$) matrices, see Dray and Manogue [13]). The dimension of $\mathcal{V}_{m,n}^{\beta}$ over $\Re$ is $[\beta mn-m(m-1)\beta/2-m]$. In particular, ${\mathcal{V}}_{m,m}^{\beta}$ with dimension over $\Re$, $[m(m+1)\beta/2-m]$, is the maximal compact subgroup $\mathfrak{U}^{\beta}(m)$ of ${\mathcal{L}}^{\beta}_{m,m}$ and consists of all matrices $\mathbf{H}\in\mathfrak{F}^{m\times m}$ such that $\mathbf{H}^{H}\mathbf{H}=\mathbf{I}_{m}$. Therefore, $\mathfrak{U}^{\beta}(m)$ is the real orthogonal group $\mathcal{O}(m)$ ($\beta=1$), the unitary group $\mathcal{U}(m)$ ($\beta=2$), compact symplectic group $\mathcal{S}p(m)$ ($\beta=4$) or exceptional type matrices $\mathcal{O}o(m)$ ($\beta=8$), for $\mathfrak{F}=\Re$, $\mathfrak{C}$, $\mathfrak{H}$ or $\mathfrak{O}$, respectively.

We denote by ${\mathfrak{S}}_{m}^{\beta}$ the real vector space of all $\mathbf{S}\in\mathfrak{F}^{m\times m}$ such that $\mathbf{S}=\mathbf{S}^{H}$. Let $\mathfrak{P}_{m}^{\beta}$ be the cone of positive definite matrices $\mathbf{S}\in\mathfrak{F}^{m\times m}$; then $\mathfrak{P}_{m}^{\beta}$ is an open subset of ${\mathfrak{S}}_{m}^{\beta}$. Over $\Re$, ${\mathfrak{S}}_{m}^{\beta}$ consist of symmetric matrices; over $\mathfrak{C}$, Hermitian matrices; over $\mathfrak{H}$, quaternionic Hermitian matrices (also termed self-dual matrices) and over $\mathfrak{O}$, octonionic Hermitian matrices. Generically, the elements of $\mathfrak{S}_{m}^{\beta}$ are termed Hermitian matrices, irrespective of the nature of $\mathfrak{F}$. The dimension of $\mathfrak{S}_{m}^{\beta}$ over $\Re$ is $[m(m-1)\beta+2m]/2$.

Let $\mathfrak{D}_{m}^{\beta}$ be the diagonal subgroup of $\mathcal{L}_{m,m}^{\beta}$ consisting of all $\mathbf{D}\in\mathfrak{F}^{m\times m}$, $\mathbf{D}=\mathop{\rm diag}\nolimits(d_{1},\dots,d_{m})$.

For any matrix $\mathbf{X}\in\mathfrak{F}^{n\times m}$, $d\mathbf{X}$ denotes the matrix of differentials $(dx_{ij})$. Finally, we define the measure or volume element $(d\mathbf{X})$ when $\mathbf{X}\in\mathfrak{F}^{m\times n},\mathfrak{S}_{m}^{\beta}$, $\mathfrak{D}_{m}^{\beta}$ or $\mathcal{V}_{m,n}^{\beta}$, see Dimitriu [12].

If $\mathbf{X}\in\mathfrak{F}^{n\times m}$ then $(d\mathbf{X})$ (the Lebesgue measure in $\mathfrak{F}^{n\times m}$) denotes the exterior product of the $\beta mn$ functionally independent variables

If $\mathbf{S}\in\mathfrak{S}_{m}^{\beta}$ (or $\mathbf{S}\in\mathfrak{T}_{L}^{\beta}(m)$) then $(d\mathbf{S})$ (the Lebesgue measure in $\mathfrak{S}_{m}^{\beta}$ or in $\mathfrak{T}_{L}^{\beta}(m)$) denotes the exterior product of the $m(m+1)\beta/2$ functionally independent variables (or denotes the exterior product of the $m(m-1)\beta/2+m$ functionally independent variables, if $s_{ii}\in\Re$ for all $i=1,\dots,m$)

The context generally establishes the conditions on the elements of $\mathbf{S}$, that is, if $s_{ij}\in\Re$, $\in\mathfrak{C}$, $\in\mathfrak{H}$ or $\in\mathfrak{O}$. It is considered that

Observe, too, that for the Lebesgue measure $(d\mathbf{S})$ defined thus, it is required that $\mathbf{S}\in\mathfrak{P}_{m}^{\beta}$, that is, $\mathbf{S}$ must be a non singular Hermitian matrix (Hermitian positive definite matrix).

If $\mathbf{\Lambda}\in\mathfrak{D}_{m}^{\beta}$ then $(d\mathbf{\Lambda})$ (the Legesgue measure in $\mathfrak{D}_{m}^{\beta}$) denotes the exterior product of the $\beta m$ functionally independent variables

If $\mathbf{H}_{1}\in\mathcal{V}_{m,n}^{\beta}$ then

where $\mathbf{H}=(\mathbf{H}_{1}|\mathbf{H}_{2})=(\mathbf{h}_{1},\dots,\mathbf{h}_{m}|\mathbf{h}_{m+1},\dots,\mathbf{h}_{n})\in\mathfrak{U}^{\beta}(m)$. It can be proved that this differential form does not depend on the choice of the $\mathbf{H}_{2}$ matrix. When $m=1$; $\mathcal{V}^{\beta}_{1,n}$ defines the unit sphere in $\mathfrak{F}^{n}$. This is, of course, an $(n-1)\beta$- dimensional surface in $\mathfrak{F}^{n}$. When $m=n$ and denoting $\mathbf{H}_{1}$ by $\mathbf{H}$, $(\mathbf{H}^{H}d\mathbf{H})$ is termed the Haar measure on $\mathfrak{U}^{\beta}(m)$.

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

where $\Gamma^{\beta}_{m}[a]$ denotes the multivariate Gamma function for the space $\mathfrak{S}_{m}^{\beta}$, and is defined by

where $\mathop{\rm etr}\nolimits(\cdot)=\exp(\mathop{\rm tr}\nolimits(\cdot))$, $|\cdot|$ denotes the determinant and $\mathop{\rm Re}\nolimits(a)>(m-1)\beta/2$, see Gross and Richards [20]. If $\mathbf{A}\in\mathcal{L}_{m,n}^{\beta}$ then by $\mathop{\rm vec}\nolimits(\mathbf{A})$ we mean the $mn\times 1$ vector formed by stacking the columns of $\mathbf{A}$ under each other; that is, if $\mathbf{A}=[\mathbf{a}_{1}\mathbf{a}_{2}\dots\mathbf{a}_{m}]$, where $\mathbf{a}_{j}\in\mathcal{L}_{1,n}^{\beta}$ for $j=1,2,\dots,m$

Below are summarised some Jacobians in terms of the $\beta$ parameter. For a detailed discussion of this and related issues see Dimitriu [12], Edelman and Rao [15], Forrester [19] and Kabe [22].

As a consequence of this result, we have the following statement.

Finally, note that for $a\in\mathcal{L}_{1,1}^{\beta}$ constant, making the change of variable $v=u/a$ and $(du)=a^{\beta}(dv)$ in Fang et al. [18, Equation 2.21, p. 26] we have

Finally, from Díaz-García, and Gutiérrez-Jáimez [10]

where $\mathbf{V}\in\mathfrak{P}_{m}^{\beta}$ and $\mathbf{\Sigma}\in\mathfrak{P}_{m}^{\beta}$.

First, note that any new particular distribution indexed by the kernel $h(\cdot)$ form part of its density function, then it defines a family of distributions itself in terms of each possible choice of $h(\cdot)$.

Assume that $\mathbf{X}\sim\mathcal{E}_{n\times m}^{\beta}(\mathbf{0},\mathbf{I}_{n},\mathbf{I}_{m};h)$, such that $n_{0}+n_{1}+\cdots+n_{k}=n$, and $\mathbf{X}=\left(\mathbf{X}^{H}_{0},\mathbf{X}^{H}_{1},\dots,\mathbf{X}^{H}_{k}\right)^{H}$. Then (14) can be written as

where $\mathbf{X}_{i}\in\mathcal{L}_{m,n_{i}}^{\beta}$, $i=0,1,\dots,k$. Take in account that, only under a matrix variate normal distribution the random matrices are independent, see Fang and Zhang [17], Gupta and Varga [21] and Fang et al. [18]. In general, the random matrices $\mathbf{X}_{0},\mathbf{X}_{1},\dots,\mathbf{X}_{k}$ are probabilistically dependent.

Proceeding as Díaz-García et al. [7, Equation 4.2], defining $V_{i}=\mathop{\rm tr}\nolimits\mathbf{X}_{i}^{H}\mathbf{X}_{i}$, $i=0,1,\dots,k$, and as in Díaz-García and Caro-Lopera [5, Equation (1), p. 216] the following result in general case, is obtained.