Hotelling's T-squared Distribution - Hotelling's Two-sample T-squared Statistic

Hotelling's Two-sample T-squared Statistic

If and, with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define

as the sample means, and

${\mathbf W}= \frac{\sum_{i=1}^{n_x}(\mathbf{x}_i-\overline{\mathbf x})(\mathbf{x}_i-\overline{\mathbf x})' +\sum_{i=1}^{n_y}(\mathbf{y}_i-\overline{\mathbf y})(\mathbf{y}_i-\overline{\mathbf y})'}{n_x+n_y-2}$

as the unbiased pooled covariance matrix estimate, then Hotelling's two-sample T-squared statistic is

$t^2 = \frac{n_x n_y}{n_x+n_y}(\overline{\mathbf x}-\overline{\mathbf y})'{\mathbf W}^{-1}(\overline{\mathbf x}-\overline{\mathbf y}) \sim T^2(p, n_x+n_y-2)$

and it can be related to the F-distribution by

The non-null distribution of this statistic is the noncentral F-distribution (the ratio of a non-central Chi-squared random variable and an independent central Chi-squared random variable)

with

where is the difference vector between the population means.

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