Matrix variate distribution

A matrix random phenomenon is an observable phenomenon that can be represented in matrix form and that, under repeated observations, yields different outcomes which are not deterministically predictable. Instead, the outcomes obey certain conditions of statistical regularity. The set of descriptions of all possible outcomes that may occur on observing a matrix random phenomenon is the sampling space $\mathcal{S}$. A matrix event is a subset of $\mathcal{S}$. A measure of the degree of certainty with which a given matrix event will occur when observing a matrix random phenomenon can be found by defining a probability function on subsets of $\mathcal{S}$, assigning a probability to every matrix event.

A matrix $X ( p \times n )$ consisting of $n p$ elements $x _ { 11 } ( \cdot ) , \ldots , x _ { p n } ( \cdot )$ which are real-valued functions defined on $\mathcal{S}$ is a real random matrix if the range $\mathbf{R} ^ { p \times n }$ of

\begin{equation*} \left( \begin{array} { c c c } { x _ { 11 } ( . ) } & { \dots } & { x _ { 1 n } ( . ) } \\ { \vdots } & { \square } & { \vdots } \\ { x _ { p 1 } ( . ) } & { \dots } & { x _ { p n } (1) } \end{array} \right), \end{equation*}

consists of Borel sets in the $n p$-dimensional real space and if for each Borel set $B$ of real $n p$-tuples, arranged in a matrix,

\begin{equation*} \left( \begin{array} { c c c } { x _ { 11 } } & { \dots } & { x _ { 1 n} } \\ { \vdots } & { \square } & { \vdots } \\ { x _ { p 1 } } & { \dots } & { x _ { p n} } \end{array} \right), \end{equation*}

in $\mathbf{R} ^ { p \times n }$, the set

\begin{equation*} \left\{ s \in \mathcal{S} : \left( \begin{array} { c c c } { x _ { 11 } ( s _ { 11 } ) } & { \dots } & { x _ { 1 n } ( s _ { 1 n } ) } \\ { \vdots } & { \square } & { \vdots } \\ { x _ { p 1 } ( s _ { p 1 } ) } & { \dots } & { x _ { p n } ( s _ { p n } ) } \end{array} \right) \in B \right\} \end{equation*}

is an event in $\mathcal{S}$. The probability density function of $X$ (cf. also Density of a probability distribution) is a scalar function $f _ { X } ( X )$ such that:

i) $f _ { X } ( X ) \geq 0$;

ii) $\int _ { X } f _ { X } ( X ) d X = 1$; and

iii) $\mathsf{P} ( X \in A ) = \int _ { A } f _ { X } ( X ) d X$, where $A$ is a subset of the space of realizations of $X$. A scalar function $f _ { X , Y } ( X , Y )$ defines the joint (bi-matrix variate) probability density function of $X$ and $Y$ if

a) $f _ { X , Y } ( X , Y ) \geq 0$;

b) $\int _ { Y } \int_X f _ { X , Y } d X d Y = 1$; and

c) $\mathsf{P} ( ( X , Y ) \in A ) = \int \int _ { A } f _ { X , Y } d X d Y$, where $A$ is a subset of the space of realizations of $( X , Y )$.

The marginal probability density function of $X$ is defined by $f _ { X } ( X ) = \int _ { Y } f _ { X , Y } ( X , Y ) d Y$, and the conditional probability density function of $X$ given $Y$ is defined by

\begin{equation*} f _ { X | Y } ( X | Y ) = \frac { f _ { X , Y } ( X , Y ) } { f _ { Y } ( Y ) } ,\; f _ { Y } ( Y ) > 0, \end{equation*}

where $f _ { Y } ( Y )$ is the marginal probability density function of $Y$.

Two random matrices $X ( p \times n )$ and $Y ( r \times s )$ are independently distributed if and only if

\begin{equation*} f _ { X , Y } ( X , Y ) = f _ { X } ( X ) f _ { Y } ( Y ), \end{equation*}

where $f _ { X } ( X )$ and $f _ { Y } ( Y )$ are the marginal densities of $X$ and $Y$, respectively.

The characteristic function of the random matrix $X ( p \times n )$ is defined as

\begin{equation*} \phi _ { X } ( Z ) = \int _ { X } \operatorname { etr } ( i Z X ^ { \prime } ) f _ { X } ( X ) d X \end{equation*}

where $Z ( p \times n )$ is a real arbitrary matrix and is the exponential trace function $\operatorname { etr } ( A ) = \operatorname { exp } ( \operatorname { tr } ( A ) )$.

For the random matrix $X ( p \times n ) = ( X _ { ij } )$, the mean matrix is given by $\mathsf{E} ( X ) = ( \mathsf{E} ( X _ { ij } ) )$. The $( p n \times r s )$ covariance matrix of the random matrices $X ( p \times n )$ and $Y ( r \times s )$ is defined by

Examples of matrix variate distributions.

The matrix variate normal distribution

\begin{equation*} \frac { 1 } { ( 2 \pi ) ^ { n p / 2 } | \Sigma | ^ { n / 2 } | \Psi | ^ { p / 2 } } \times \end{equation*}

\begin{equation*} \times \operatorname { etr } \left\{ - \frac { 1 } { 2 } \Sigma ^ { - 1 } ( X - M ) \Psi ^ { - 1 } ( X - M ) ^ { \prime } \right\} , X \in \mathbf{R} ^ { p \times n } , M \in \mathbf{R} ^ { p \times n } , \Sigma > 0 , \Psi > 0. \end{equation*}

The Wishart distribution

\begin{equation*} \frac { 1 } { 2 ^ { n p / 2 } \Gamma _ { p } ( n / 2 ) | \Sigma | ^ { n / 2 } } | S | ^ { ( n - p - 1 ) / 2 } \operatorname { etr } \left( - \frac { 1 } { 2 } \Sigma ^ { - 1 } S \right), \end{equation*}

\begin{equation*} S > 0 , n \geq p. \end{equation*}

The matrix variate $t$-distribution

\begin{equation*} \frac { \Gamma _ { p } \left[ \frac { \langle n + m + p - 1 \rangle} { 2 } \right] } { \pi ^ { m p / 2 } \Gamma _ { p } ( ( n + p - 1 ) / 2 ) } | \Sigma | ^ { - m / 2 } | \Omega | ^ { - p / 2 } \times \end{equation*}

\begin{equation*} \times \left| I _ { p } + \Sigma ^ { - 1 } ( X - M ) \Omega ^ { - 1 } ( X - M ) ^ { \prime } \right| ^ { - ( n + m + p - 1 ) / 2 } , X \in {\bf R} ^ { p \times n } , M \in {\bf R} ^ { p \times n } , \Sigma > 0 , \Omega > 0. \end{equation*}

The matrix variate beta-type-I distribution

\begin{equation*} \frac { 1 } { \beta _ { p } ( a , b ) } | U | ^ { a - ( p + 1 ) / 2 } | I _ { p } - U | ^ { b - ( p + 1 ) / 2 }, \end{equation*}

\begin{equation*} 0 < U < I _ { p } , a > \frac { 1 } { 2 } ( p - 1 ) , b > \frac { 1 } { 2 } ( p - 1 ). \end{equation*}

The matrix variate beta-type-II distribution

\begin{equation*} \frac { 1 } { \beta _ { p } ( a , b ) } | V | ^ { a - ( p + 1 ) / 2 } | I _ { p } + V | ^ { - ( a + b ) }, \end{equation*}

\begin{equation*} V > 0 , a > \frac { 1 } { 2 } ( p - 1 ) , b > \frac { 1 } { 2 } ( p - 1 ). \end{equation*}

References

[a1]	P. Bougerol, J. Lacroix, "Products of random matrices with applications to Schrödinger operators" , Birkhäuser (1985)
[a2]	M. Carmeli, "Statistical theory and random matrices" , M. Dekker (1983)
[a3]	"Random matrices and their applications" J.E. Cohen (ed.) H. Kesten (ed.) C.M. Newman (ed.) , Amer. Math. Soc. (1986)
[a4]	A.K. Gupta, T. Varga, "Elliptically contoured models in statistics" , Kluwer Acad. Publ. (1993)
[a5]	A.K. Gupta, V.L. Girko, "Multidimensional statistical analysis and theory of random matrices" , VSP (1996)
[a6]	M.L. Mehta, "Random matrices" , Acad. Press (1991) (Edition: Second)