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 follows. Suppose

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". (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 t² is closely related to the squared Mahalanobis distance.

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 t² is T²(p,n), Hotelling's T-square distribution with parameters p and n.

The assumptions above are frequently met in practice: it can be shown ^[2] that if , are independent, and and are as defined above then is independent of , and

.

If, moreover, both distributions are nonsingular, it can be shown^[2] that

and

where F is the F-distribution.

[edit] 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

^[2]

[edit] See also

• Student's t-distribution (the univariate equivalent)
• F-distribution (commonly tabulated or available in software libraries, and hence used for testing the T-square statistic using the relationship given above)
• Wilks' lambda distribution (in multivariate statistics Wilks' Λ is to Hotelling's T² as Snedecor's F is to Student's t in univariate statistics).

[edit] References

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

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