Hotelling's T-squared distribution
The Hotelling's T-square distribution is a probability distribution, which was described in 1931 by Harold Hotelling first time. It is a generalization of the Student's t-distribution.
Hotelling's T -square distribution is defined as
- A number of points
- Is a column vector with elements
- Is a covariance matrix.
It is a random variable with a multivariate normal distribution and ( independent of x ) have a Wishart distribution with a non- singular variance matrix and with. Then the distribution of: Hotelling's T -square distribution with parameters p and m.
Is the F- distribution. Then it can be shown that:
P × 1 column vectors with real numbers.
Is the mean value. The positive definite matrix P × P
Is their "sample variance " matrix. ( The transpose of a matrix M is M 'denotes ). μ is a P × 1 column vector (using an estimate of the mean value ). Then the Hotelling's T -square distribution
Has a close relationship with the squared Mahalanobis distance.
In particular, it can be shown that if and are independently and are defined as above, then has a Wishart distribution with n-1 degrees of freedom