Hotelling's T-square distribution

In statistics, Hotelling's T-square statistic, 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

${\mathbf x}_1,\dots,{\mathbf x}_n$

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

$\overline{\mathbf x}=(\mathbf{x}_1+\cdots+\mathbf{x}_n)/n$

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

${\mathbf W}=\sum_{i=1}^n (\mathbf{x}_i-\overline{\mathbf x})(\mathbf{x}_i-\overline{\mathbf x})'/(n-1)$

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

$T^2=n(\overline{\mathbf x}-{\mathbf\mu})'{\mathbf W}^{-1}(\overline{\mathbf x}-{\mathbf\mu}).$

Note that $T 2$ may be determined for any matrix of rank at least p.

The reason that this is interesting is that if $\mathbf{x}\sim N_p(\mu,{\mathbf V})$ is a random variable with a multivariate normal distribution and ${\mathbf W}\sim W_p(n,{\mathbf V})$ has a Wishart distribution, and ${\mathbf x}$ and ${\mathbf W}$ are independent, then the probability distribution of $T 2$ is Hotelling's T-square distribution.

The assumptions above are frequently met in practice: it can be shown that if ${\mathbf x}_1,\dots,{\mathbf x}_n\sim N_p(\mu,{\mathbf V})$ , are independent, and $\overline{\mathbf x}$ and ${\mathbf W}$ are as defined above then ${\mathbf W}$ has a Wishart distribution with m = n − 1 degrees of freedom and is independent of $\overline{\mathbf x}$ , and

$\overline{\mathbf x}\sim N_p(\mu,V/n).$

If, moreover, both distributions are nonsingular, it can be shown that

$\frac{m-p+1}{pm} T^2\sim F_{p,m-p+1}$

where $F$ is the F-distribution.

Categories: Probability distributions