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 . A matrix event is a subset of . 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 , assigning a probability to every matrix event.

A matrix consisting of elements which are real-valued functions defined on is a real random matrix if the range of

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

in , the set

is an event in . The probability density function of (cf. also Density of a probability distribution) is a scalar function such that:

i) ;

ii) ; and

iii) , where is a subset of the space of realizations of . A scalar function defines the joint (bi-matrix variate) probability density function of and if

a) ;

b) ; and

c) , where is a subset of the space of realizations of .

The marginal probability density function of is defined by , and the conditional probability density function of given is defined by

where is the marginal probability density function of .

Two random matrices and are independently distributed if and only if

where and are the marginal densities of and , respectively.

The characteristic function of the random matrix is defined as

where is a real arbitrary matrix and is the exponential trace function .

For the random matrix , the mean matrix is given by . The covariance matrix of the random matrices and is defined by

Examples of matrix variate distributions.

The matrix variate normal distribution

The Wishart distribution

The matrix variate -distribution

The matrix variate beta-type-I distribution

The matrix variate beta-type-II distribution

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)