<s>
In	O
statistics	O
,	O
the	O
multivariate	B-General_Concept
t-distribution	I-General_Concept
(	O
or	O
multivariate	B-General_Concept
Student	I-General_Concept
distribution	I-General_Concept
)	O
is	O
a	O
multivariate	O
probability	O
distribution	O
.	O
</s>
<s>
It	O
is	O
a	O
generalization	O
to	O
random	B-General_Concept
vectors	I-General_Concept
of	O
the	O
Student	O
's	O
t-distribution	O
,	O
which	O
is	O
a	O
distribution	O
applicable	O
to	O
univariate	O
random	O
variables	O
.	O
</s>
<s>
While	O
the	O
case	O
of	O
a	O
random	O
matrix	B-Architecture
could	O
be	O
treated	O
within	O
this	O
structure	O
,	O
the	O
matrix	B-General_Concept
t-distribution	I-General_Concept
is	O
distinct	O
and	O
makes	O
particular	O
use	O
of	O
the	O
matrix	B-Architecture
structure	O
.	O
</s>
<s>
One	O
common	O
method	O
of	O
construction	O
of	O
a	O
multivariate	B-General_Concept
t-distribution	I-General_Concept
,	O
for	O
the	O
case	O
of	O
dimensions	O
,	O
is	O
based	O
on	O
the	O
observation	O
that	O
if	O
and	O
are	O
independent	O
and	O
distributed	O
as	O
and	O
(	O
i.e.	O
</s>
<s>
and	O
is	O
said	O
to	O
be	O
distributed	O
as	O
a	O
multivariate	B-General_Concept
t-distribution	I-General_Concept
with	O
parameters	O
.	O
</s>
<s>
Note	O
that	O
is	O
not	O
the	O
covariance	O
matrix	B-Architecture
since	O
the	O
covariance	O
is	O
given	O
by	O
(	O
for	O
)	O
.	O
</s>
<s>
The	O
constructive	O
definition	O
of	O
a	O
multivariate	B-General_Concept
t-distribution	I-General_Concept
simultaneously	O
serves	O
as	O
a	O
sampling	O
algorithm	O
:	O
</s>
<s>
This	O
formulation	O
gives	O
rise	O
to	O
the	O
hierarchical	O
representation	O
of	O
a	O
multivariate	B-General_Concept
t-distribution	I-General_Concept
as	O
a	O
scale-mixture	O
of	O
normals	O
:	O
where	O
indicates	O
a	O
gamma	O
distribution	O
with	O
density	O
proportional	O
to	O
,	O
and	O
conditionally	O
follows	O
.	O
</s>
<s>
The	O
definition	O
of	O
the	O
cumulative	O
distribution	O
function	O
(	O
cdf	O
)	O
in	O
one	O
dimension	O
can	O
be	O
extended	O
to	O
multiple	O
dimensions	O
by	O
defining	O
the	O
following	O
probability	O
(	O
here	O
is	O
a	O
real	O
vector	B-General_Concept
)	O
:	O
</s>
<s>
There	O
is	O
no	O
simple	O
formula	O
for	O
,	O
but	O
it	O
can	O
be	O
via	O
Monte	B-Algorithm
Carlo	I-Algorithm
integration	I-Algorithm
.	O
</s>
<s>
Let	O
vector	B-General_Concept
follow	O
the	O
multivariate	B-General_Concept
t	I-General_Concept
distribution	I-General_Concept
and	O
partition	O
into	O
two	O
subvectors	O
of	O
elements	O
:	O
</s>
<s>
where	O
,	O
the	O
known	O
mean	O
vector	B-General_Concept
is	O
and	O
the	O
scale	O
matrix	B-Architecture
is	O
.	O
</s>
<s>
Kibria	O
and	O
Joarder	O
,	O
in	O
a	O
tutorial-style	O
paper	O
,	O
define	O
radial	O
measure	O
such	O
thatwhich	O
is	O
equivalent	O
to	O
the	O
expected	O
variance	O
of	O
-element	O
vector	B-General_Concept
treated	O
as	O
a	O
univariate	O
zero-mean	O
random	O
sequence	O
.	O
</s>
<s>
They	O
note	O
that	O
follows	O
the	O
Fisher-Snedecor	B-General_Concept
or	O
distribution	O
:	O
</s>
<s>
To	O
scale	O
the	O
radial	O
variables	O
without	O
changing	O
the	O
radial	O
shape	O
function	O
,	O
define	O
scale	O
matrix	B-Architecture
,	O
yielding	O
a	O
3-parameter	O
Cartesian	O
density	O
function	O
,	O
ie	O
.	O
</s>
<s>
let	O
be	O
a	O
-vector	O
sampled	O
from	O
a	O
central	O
spherical	O
multivariate	B-General_Concept
t	I-General_Concept
distribution	I-General_Concept
with	O
degrees	O
of	O
freedom	O
:	O
.	O
</s>
<s>
Roth	O
(	O
reference	O
below	O
)	O
notes	O
that	O
if	O
is	O
a	O
squat	O
matrix	B-Architecture
with	O
then	O
has	O
distribution	O
.	O
</s>
<s>
In	O
the	O
above	O
,	O
the	O
degrees	O
of	O
freedom	O
parameter	O
remains	O
invariant	O
throughout	O
and	O
all	O
vectors	O
must	O
ultimately	O
derive	O
from	O
one	O
initial	O
isotropic	O
spherical	O
vector	B-General_Concept
whose	O
elements	O
are	O
not	O
statistically	O
independent	O
.	O
</s>
<s>
Adding	O
two	O
sample	O
multivariate	O
t	O
vectors	O
generated	O
with	O
independent	O
Chi-squared	O
samples	O
and	O
different	O
values	O
:	O
,	O
as	O
defined	O
in	O
the	O
leading	O
paragraph	O
,	O
will	O
not	O
produce	O
internally	O
consistent	O
distributions	O
,	O
though	O
they	O
will	O
yield	O
a	O
Behrens-Fisher	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
In	O
univariate	O
statistics	O
,	O
the	O
Student	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
makes	O
use	O
of	O
Student	O
's	O
t-distribution	O
.	O
</s>
<s>
The	O
matrix	B-General_Concept
t-distribution	I-General_Concept
is	O
a	O
distribution	O
for	O
random	O
variables	O
arranged	O
in	O
a	O
matrix	B-Architecture
structure	O
.	O
</s>
