Matrix variate elliptically contoured distribution

The class of matrix variate elliptically contoured distributions can be defined in many ways. Here the definition of A.K. Gupta and T. Varga [a4] is given.

A random matrix $X ( p \times n )$ (see Matrix variate distribution) is said to have a matrix variate elliptically contoured distribution if its characteristic function has the form $\phi _ { X } ( T ) = \operatorname { etr } ( i T ^ { \prime } M ) \psi ( \operatorname { tr } ( T ^ { \prime } \Sigma T \Phi ) )$ with $T$ a $( p \times n )$-matrix, $M$ a $( p \times n )$-matrix, $\Sigma$ a $( p \times p )$-matrix, $\Phi$ a $( n \times n )$-matrix, $\Sigma \geq 0$, $\Phi \geq 0$ and $\psi : [ 0 , \infty ) \rightarrow \mathbf{R}$. This distribution is denoted by $E _ { p , n } ( M , \Sigma \otimes \Phi , \psi )$. If the distribution of $X$ is absolutely continuous (cf. also Absolute continuity), then its probability density function (cf. also Density of a probability distribution) has the form

\begin{equation*} | \Sigma | ^ { - n / 2 } | \Phi | ^ { - p / 2 } h ( \operatorname { tr } \left( ( X - M ) ^ { \prime } \Sigma ^ { - 1 } ( X - M ) \Phi ^ { - 1 } ) \right), \end{equation*}

where $h$ and $\psi$ determine each other.

An important subclass of the class of matrix variate elliptically contoured distributions is the class of matrix variate normal distributions. A matrix variate elliptically contoured distribution has many properties which are similar to the normal distribution. For example, linear functions of a random matrix with a matrix variate elliptically contoured distribution also have elliptically contoured distributions. That is, if $X \sim E _ { p , n } ( M , \Sigma \otimes \Phi , \psi )$, then for given constant matrices $C ( q \times n )$, $A ( q \times p )$, $B ( n \times m )$, $A X B + C \sim E _ { q , n } ( A M B + C , ( A \Sigma A ^ { \prime } ) \otimes ( B ^ { \prime } \Phi B ) , \psi )$.

If $X$, $M$ and $\Sigma$ are partitioned as

\begin{equation*} X = \left( \begin{array} { l } { X _ { 1 } } \\ { X _ { 2 } } \end{array} \right) , M = \left( \begin{array} { c } { M _ { 1 } } \\ { M _ { 2 } } \end{array} \right) , \Sigma = \left( \begin{array} { l l } { \Sigma _ { 11 } } & { \Sigma _ { 12 } } \\ { \Sigma _ { 21 } } & { \Sigma _ { 22 } } \end{array} \right), \end{equation*}

where $X _ { 1 }$ is a $( q \times n )$-matrix, $M _ { 1 }$ is a $( q \times n )$-matrix and $\Sigma _ { 11 }$ is a $( q \times q )$-matrix, $q < p$, then $X _ { 1 } \sim E _ { q ,\, n } ( M _ { 1 } , \Sigma _ { 11 } \otimes \Phi , \psi )$. However, if $X$, $M$ and $\Phi$ are partitioned as

\begin{equation*} X = ( X _ { 1 } , X _ { 2 } ) , M = ( M _ { 1 } , M _ { 2 } ) , \Phi = \left( \begin{array} { c c } { \Phi _ { 11 } } & { \Phi _ { 12 } } \\ { \Phi _ { 21 } } & { \Phi _ { 22 } } \end{array} \right), \end{equation*}

where $X _ { 1 }$ is a $( p \times m )$-matrix, $M _ { 1 }$ is a $( p \times m )$-matrix, and $\Phi _ { 11 }$ is an $( m \times m )$-matrix, $m < n$, then $X _ { 1 } \sim E _ { p , m } ( M _ { 1 } , \Sigma \otimes \Phi _ { 11 } , \psi )$.

Here, if the expectations exist, then $\mathsf{E} ( X ) = M$ and $\operatorname { cov } ( X ) = c \Sigma \otimes \Phi$, where $c = - 2 \psi ^ { \prime } ( 0 )$. An important tool in the study of matrix variate elliptically contoured distributions is the stochastic representation of $X$:

\begin{equation*} X : = M + r A U B ^ { \prime }, \end{equation*}

where $\operatorname{rank} ( \Sigma ) = p _ { 1 }$, $\operatorname{rank} ( \Phi ) = n _ { 1 }$, $U$ is a $( p _ { 1 } \times n _ { 1 } )$-matrix and $\overset{\rightharpoonup} { f n n m e } ( U ^ { \prime } )$ is uniformly distributed on the unit sphere in $\mathbf{R} ^ { p_1 n_1 } $, $r$ is a non-negative random variable, $r$ and $U$ are independent, $\Sigma = A A ^ { \prime }$, and $\Phi = B B ^ { \prime }$. Moreover,

\begin{equation*} \psi ( u ) = \int _ { 0 } ^ { \infty } \Omega _ { p _ { 1 } n _ { 1 } } ( r ^ { 2 } u ) d F ( r ) , u \geq 0, \end{equation*}

where $\Omega _ { p _ { 1 } n _ { 1 } } ( t ^ { \prime } t ^ { \prime } )$, $t \in {\bf R} ^ { p _ { 1 } n _ { 1 } }$ denotes the characteristic function of $\overset{\rightharpoonup} { f n n m e } ( U ^ { \prime } )$, and $F ( r )$ denotes the distribution function of $r$.

References

[a1]	K.T. Fang, Y.T. Zhang, "Generalized multivariate analysis" , Springer (1990)
[a2]	K.T. Fang, T.W. Anderson, "Statistical inference in elliptically contoured and related distributions" , Allerton Press (1990)
[a3]	A.K. Gupta, T. Varga, "Rank of a quadratic form in an elliptically contoured matrix random variable" Statist. Probab. Lett. , 12 (1991) pp. 131–134
[a4]	A.K. Gupta, T. Varga, "Elliptically contoured models in statistics" , Kluwer Acad. Publ. (1993)
[a5]	A.K. Gupta, T. Varga, "Some applications of the stochastic representation of elliptically contoured distribution" Random Oper. and Stoch. Eqs. , 2 (1994) pp. 1–11
[a6]	A.K. Gupta, T. Varga, "A new class of matrix variate elliptically contoured distributions" J. Italian Statist. Soc. , 3 (1994) pp. 255–270
[a7]	A.K. Gupta, T. Varga, "Moments and other expected values for matrix variate elliptically contoured distributions" Statistica , 54 (1994) pp. 361–373
[a8]	A.K. Gupta, T. Varga, "Normal mixture representation of matrix variate elliptically contoured distributions" Sankhyā Ser. A , 57 (1995) pp. 68–78
[a9]	A.K. Gupta, T. Varga, "Some inference problems for matrix variate elliptically contoured distributions" Statistics , 26 (1995) pp. 219–229
[a10]	A.K. Gupta, T. Varga, "Characterization of matrix variate elliptically contoured distributions" , Adv. Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz , Wiley (1997) pp. 455–467