Hotelling's T^2 Test

Hotelling's test is a multivariate generalization of Student's -test. For a random sample from a -dimensional multivariate normal distribution, let be the sample mean and let be the unbiased sample covariance matrix

(1)

To test the null hypothesis that the population mean is , the one-sample statistic is

(2)

When the null hypothesis holds and ,

(3)

For two independent multivariate normal samples of sizes and having a common covariance matrix, let and be their unbiased sample covariance matrices, and define the pooled sample covariance matrix by

(4)

The statistic for testing equality of the two population mean vectors is

(5)

and under the null hypothesis,

(6)

Thus both forms can be evaluated using the F-distribution. The test requires a nonsingular estimated covariance matrix and is therefore unsuitable without modification when the dimension is too large relative to the sample size.