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Hotelling's T-square DistributionIn 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 -
T^2=n(\overline{\mathbf x}-{\mathbf\mu})'{\mathbf W}^{-1}(\overline{\mathbf x}-{\mathbf\mu}). Note that may be determined for any matrix of rank at least p. The reason that this is interesting is that 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. The assumptions above are frequently met in practice: 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 -
\frac{m-p+1}{pm} T^2\sim F_{p,m-p+1} where is the F-distribution.
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