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The continuous probability distribution, concentrated on the positive semi-axis , with density

depending on two integer parameters (the number of degrees of freedom) and , . For the Hotelling -distribution reduces to the Student distribution, and for any it can be regarded as a multivariate generalization of the Student distribution in the following sense. If a -dimensional random vector has the normal distribution with null vector of means and covariance matrix and if

where the random vectors are independent, distributed as and also independent of , then the random variable has the Hotelling -distribution with degrees of freedom ( is a column vector and means transposition). If , then

where the random variable has the Student distribution with degrees of freedom. If in the definition of the random variable it is assumed that has the normal distribution with parameters and has the normal distribution with parameters , then the corresponding distribution is called a non-central Hotelling -distribution with degrees of freedom and non-centrality parameter .

Hotelling's -distribution is used in mathematical statistics in the same situation as Student's -distribution, but then in the multivariate case (see Multi-dimensional statistical analysis). If the results of observations are independent normally-distributed random vectors with mean vector and non-degenerate covariance matrix , then the statistic



has the Hotelling -distribution with degrees of freedom. This fact forms the basis of the Hotelling test. For numerical calculations one uses tables of the beta-distribution or of the Fisher -distribution, because the random variable has the -distribution with and degrees of freedom.

The Hotelling -distribution was proposed by H. Hotelling [1] for testing equality of means of two normal populations.


[1] H. Hotelling, "The generalization of Student's ratio" Ann. Math. Stat. , 2 (1931) pp. 360–378
[2] T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1984)
How to Cite This Entry:
Hotelling-T^2-distribution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article