<s>
In	O
statistics	O
,	O
particularly	O
in	O
hypothesis	O
testing	O
,	O
the	O
Hotelling	O
's	O
T-squared	O
distribution	O
(	O
T2	O
)	O
,	O
proposed	O
by	O
Harold	O
Hotelling	O
,	O
is	O
a	O
multivariate	B-General_Concept
probability	O
distribution	O
that	O
is	O
tightly	O
related	O
to	O
the	O
F-distribution	B-General_Concept
and	O
is	O
most	O
notable	O
for	O
arising	O
as	O
the	O
distribution	O
of	O
a	O
set	O
of	O
sample	O
statistics	O
that	O
are	O
natural	O
generalizations	O
of	O
the	O
statistics	O
underlying	O
the	O
Student	O
's	O
t-distribution	O
.	O
</s>
<s>
The	O
Hotelling	O
's	O
t-squared	O
statistic	O
(	O
t2	O
)	O
is	O
a	O
generalization	O
of	O
Student	O
's	O
t-statistic	O
that	O
is	O
used	O
in	O
multivariate	B-General_Concept
hypothesis	O
testing	O
.	O
</s>
<s>
The	O
distribution	O
arises	O
in	O
multivariate	B-General_Concept
statistics	I-General_Concept
in	O
undertaking	O
tests	O
of	O
the	O
differences	O
between	O
the	O
(	O
multivariate	B-General_Concept
)	O
means	O
of	O
different	O
populations	O
,	O
where	O
tests	O
for	O
univariate	O
problems	O
would	O
make	O
use	O
of	O
a	O
t-test	B-General_Concept
.	O
</s>
<s>
If	O
the	O
vector	O
is	O
Gaussian	O
multivariate-distributed	O
with	O
zero	O
mean	O
and	O
unit	O
covariance	O
matrix	O
and	O
is	O
a	O
matrix	O
with	O
unit	O
scale	B-Algorithm
matrix	I-Algorithm
and	O
m	O
degrees	O
of	O
freedom	O
with	O
a	O
Wishart	O
distribution	O
,	O
then	O
the	O
quadratic	O
form	O
has	O
a	O
Hotelling	O
distribution	O
(	O
with	O
parameters	O
and	O
)	O
:	O
</s>
<s>
where	O
is	O
the	O
F-distribution	B-General_Concept
with	O
parameters	O
p	O
and	O
m−p+1	O
.	O
</s>
<s>
where	O
we	O
denote	O
transpose	O
by	O
an	O
apostrophe	B-Language
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
is	O
a	O
positive	O
(	O
semi	O
)	O
definite	B-Algorithm
matrix	I-Algorithm
and	O
follows	O
a	O
p-variate	O
Wishart	O
distribution	O
with	O
n−1	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
where	O
is	O
the	O
F-distribution	B-General_Concept
with	O
parameters	O
p	O
and	O
n−p	O
.	O
</s>
<s>
Then	O
,	O
use	O
the	O
quantity	O
on	O
the	O
left	O
hand	O
side	O
to	O
evaluate	O
the	O
p-value	B-General_Concept
corresponding	O
to	O
the	O
sample	O
,	O
which	O
comes	O
from	O
the	O
F-distribution	B-General_Concept
.	O
</s>
<s>
is	O
the	O
unbiased	O
pooled	O
covariance	O
matrix	O
estimate	O
(	O
an	O
extension	O
of	O
pooled	B-General_Concept
variance	I-General_Concept
)	O
.	O
</s>
<s>
A	O
univariate	O
special	O
case	O
can	O
be	O
found	O
in	O
Welch	B-General_Concept
's	I-General_Concept
t-test	I-General_Concept
.	O
</s>
