View source for 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 [[#References|[a4]]] is given.

A random matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m1201601.png" /> (see [[Matrix variate distribution|Matrix variate distribution]]) is said to have a matrix variate elliptically contoured distribution if its [[Characteristic function|characteristic function]] has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m1201602.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m1201603.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m1201604.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m1201605.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m1201606.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m1201607.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m1201608.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m1201609.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016010.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016013.png" />. This distribution is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016014.png" />. If the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016015.png" /> is absolutely continuous (cf. also [[Absolute continuity|Absolute continuity]]), then its probability density function (cf. also [[Density of a probability distribution|Density of a probability distribution]]) has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016016.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016018.png" /> 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016019.png" />, then for given constant matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016023.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016026.png" /> are partitioned as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016028.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016029.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016030.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016031.png" />-matrix and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016032.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016033.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016035.png" />. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016038.png" /> are partitioned as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016039.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016040.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016041.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016042.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016043.png" />-matrix, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016044.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016045.png" />-matrix, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016047.png" />.

Here, if the expectations exist, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016050.png" />. An important tool in the study of matrix variate elliptically contoured distributions is the stochastic representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016051.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016052.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016055.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016056.png" />-matrix and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016057.png" /> is uniformly distributed on the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016059.png" /> is a non-negative [[Random variable|random variable]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016061.png" /> are independent, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016062.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016063.png" />. Moreover,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016064.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016066.png" /> denotes the characteristic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016067.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016068.png" /> denotes the distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120160/m12016069.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.T. Fang,   Y.T. Zhang,   "Generalized multivariate analysis" , Springer  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K.T. Fang,   T.W. Anderson,   "Statistical inference in elliptically contoured and related distributions" , Allerton Press  (1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.K. Gupta,   T. Varga,   "Elliptically contoured models in statistics" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.K. Gupta,   T. Varga,   "A new class of matrix variate elliptically contoured distributions"  ''J. Italian Statist. Soc.'' , '''3'''  (1994)  pp. 255–270</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.K. Gupta,   T. Varga,   "Moments and other expected values for matrix variate elliptically contoured distributions"  ''Statistica'' , '''54'''  (1994)  pp. 361–373</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.K. Gupta,   T. Varga,   "Normal mixture representation of matrix variate elliptically contoured distributions"  ''Sankhyā Ser. A'' , '''57'''  (1995)  pp. 68–78</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.K. Gupta,   T. Varga,   "Some inference problems for matrix variate elliptically contoured distributions"  ''Statistics'' , '''26'''  (1995)  pp. 219–229</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  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</TD></TR></table>