﻿ Hotelling's T-squared distribution

# 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.

## Definition

Hotelling's T -square distribution is defined as

With

• A number of points
• Is a column vector with elements
• Is a covariance matrix.

## Properties

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:

Assuming 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

And

It follows

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