Hotelling's T-square distribution

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In statistics, Hotelling's T-square statistic,[1] 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

where n is a number of points (see below), is a column vector of elements and is a matrix.

If is a random variable with a multivariate Gaussian distribution and (independent of x) has a Wishart distribution with the same non-singular variance matrix and with , then the distribution of is , Hotelling's T-square distribution with parameters p and m. It can be shown that

where is the F-distribution.

Now suppose that

are p×1 column vectors whose entries are real numbers. Let

be their mean. Let the p×p positive-definite matrix

be their "sample variance" matrix. (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

Note that is closely related to the squared Mahalanobis distance.

In particular, it can be shown [2] that if , are independent, and and are as defined above then has a Wishart distribution with n − 1 degrees of freedom


and is independent of , and

This implies that:

Hotelling's two-sample T-square statistic

If and , with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define

as the sample means, and

as the unbiased pooled covariance matrix estimate, then Hotelling's two-sample T-square statistic is

and it can be related to the F-distribution by


See also


  1. H. Hotelling (1931) The generalization of Student's ratio, Ann. Math. Statist., Vol. 2, pp360-378.
  2. 2.0 2.1 K.V. Mardia, J.T. Kent, and J.M. Bibby (1979) Multivariate Analysis, Academic Press.

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