multivariate probability distribution related to the F-distribution; the distribution of a set of sample statistics that are generalizations of the statistics underlying the Student's t-distribution
In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T2), proposed by Harold Hotelling, is a multivariate probability distribution that is tightly related to the F-distribution and is most notable for arising as the distribution of a set of sample statistics that are natural generalizations of the statistics underlying the Student's t-distribution.
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In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T2), proposed by Harold Hotelling, is a multivariate probability distribution that is tightly related to the F-distribution and is most notable for arising as the distribution of a set of sample statistics that are natural generalizations of the statistics underlying the Student's t-distribution.
The Hotelling's t-squared statistic (t2) is a generalization of Student's t-statistic that is used in multivariate hypothesis testing.
Contents
Motivation
The distribution arises in multivariate statistics in undertaking tests of the differences between the (multivariate) means of different populations, where tests for univariate problems would make use of a t-test.
The distribution is named for Harold Hotelling, who developed it as a generalization of Student's t-distribution.
Definition
If the vector
d
{\displaystyle d}
is Gaussian multivariate-distributed with zero mean and unit covariance matrix
Then, use the quantity on the left hand side to evaluate the p-value corresponding to the sample, which comes from the F-distribution. A confidence region may also be determined using similar logic.
be n independent identically distributed (iid) random variables, which may be represented as
The non-null distribution of this statistic is the noncentral F-distribution (the ratio of a non-central Chi-squared random variable and an independent central Chi-squared random variable)
becomes more positive. If the differences are of opposite sign
t
2
{\displaystyle t^{2}}
becomes larger as
ρ
{\displaystyle \rho }
becomes more positive.
A univariate special case can be found in Welch's t-test.
More robust and powerful tests than the Hotelling's two-sample test have been proposed in the literature, see for example the interpoint distance based tests which can be applied also when the number of variables is comparable with, or even larger than, the number of subjects.
