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 (see Matrix variate distribution) is said to have a matrix variate elliptically contoured distribution if its characteristic function has the form with a -matrix, a -matrix, a -matrix, a -matrix, , and . This distribution is denoted by . If the distribution of is absolutely continuous (cf. also Absolute continuity), then its probability density function (cf. also Density of a probability distribution) has the form

where and 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 , then for given constant matrices , , , .

If , and are partitioned as

where is a -matrix, is a -matrix and is a -matrix, , then . However, if , and are partitioned as

where is a -matrix, is a -matrix, and is an -matrix, , then .

Here, if the expectations exist, then and , where . An important tool in the study of matrix variate elliptically contoured distributions is the stochastic representation of :

where , , is a -matrix and is uniformly distributed on the unit sphere in , is a non-negative random variable, and are independent, , and . Moreover,

where , denotes the characteristic function of , and denotes the distribution function of .

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