Hotelling's T-square distribution

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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

<math>{\mathbf x}_1,\dots,{\mathbf x}_n</math>

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

<math>\overline{\mathbf x}=(\mathbf{x}_1+\cdots+\mathbf{x}_n)/n</math>

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

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

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

<math>

T^2=n(\overline{\mathbf x}-{\mathbf\mu})'{\mathbf W}^{-1}(\overline{\mathbf x}-{\mathbf\mu}). </math>

Note that <math>T^2</math> may be determined for any matrix of rank at least p.

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

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

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

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

<math>

\frac{m-p+1}{pm} T^2\sim F_{p,m-p+1} </math> where <math>F</math> is the F-distribution.it:Variabile casuale T-quadrato di Hotelling nl:Hotelling T-kwadraat