Hotelling's T-square distribution Guide, Meaning , Facts, Information and Description
In
statistics,
Hotelling's T-square statistic, named for
Harold Hotelling,
is a generalization of
Student's t statistic that is used in multivariate hypothesis testing.
Hotelling's T-square statistic is defined as follows. Suppose
are
p×1 column vectors whose entries are
real numbers. Let
be their mean. Let the
p×
p nonnegative-definite matrix
be their "sample variance". (The transpose of any matrix
M is denoted above by
M′.) Let μ be some known
p×1 column vector (in applications a hypothesized value of a population mean). Then Hotelling's T-square statistic is
If is a
random variable with a
multivariate normal distribution and has a
Wishart distribution, and and are
independent, then the
probability distribution of is
Hotelling's T-square distribution.
It can be shown that if , are independent, and and are as defined above then has a Wishart distribution with
m =
n − 1 degrees of freedom and is independent of , and
If, moreover, both distributions are nonsingular, it can be shown that
-
where is the
F-distribution.
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