The following considers a test for equality of the population mean vectors when the variance-covariance matrices are not equal.

Here we will consider the modified Hotelling's T-square test statistic given in the expression below:

\(T^2 = \mathbf{(\bar{x}_1-\bar{x}_2)}'\left\{\dfrac{1}{n_1}\mathbf{S}_1+\dfrac{1}{n_2}\mathbf{S}_2\right\}^{-1}\mathbf{(\bar{x}_1-\bar{x}_2)}\)

Again, this is a function of the differences between the sample means for the two populations. Instead of being a function of the pooled variance-covariance matrix, we can see that the modified test statistic is written as a function of the sample variance-covariance matrix, \(\mathbf{S}_{1}\), for the first population and the sample variance-covariance matrix, \(\mathbf{S}_{2}\), for the second population. It is also a function of the sample sizes \(n_{1}\) and \(n_{2}\).

For large samples, that is if both samples are large, \(T^{2}\) is approximately chi-square distributed with p d.f. We will reject \(H_{0}\) : \(\boldsymbol{\mu}_{1}\) = \(\boldsymbol{\mu}_{2}\) at level \(α\) if \(T^{2}\) exceeds the critical value from the chi-square table with p d.f. evaluated at level \(α\).

\(T^2 > \chi^2_{p, \alpha}\)

For small samples, we can calculate an F transformation as before using the formula below.

\(F = \dfrac{n_1+n_2-p-1}{p(n_1+n_2-2)}T^2\textbf{ } \overset{\cdot}{\sim}\textbf{ } F_{p,\nu}\)

This formula is a function of sample sizes \(n_{1}\) and \(n_{2}\), and the number of variables p. Under the null hypothesis this will be F-distributed with p and approximately ν degrees of freedom, where 1 divided by ν is given by the formula below:

\( \dfrac{1}{\nu} = \sum_{i=1}^{2}\frac{1}{n_i-1} \left\{ \dfrac{\mathbf{(\bar{x}_1-\bar{x}_2)}'\mathbf{S}_T^{-1}(\dfrac{1}{n_i}\mathbf{S}_i)\mathbf{S}_T^{-1}\mathbf{(\bar{x}_1-\bar{x}_2)}}{T^2} \right\} ^2 \)

This involves summing over the two samples of bank notes, a function of the number of observations of each sample, the difference in the sample mean vectors, the sample variance-covariance matrix for each of the individual samples, as well as a new matrix \(\mathbf{S}_{T}\) which is given by the expression below:

\(\mathbf{S_T} = \dfrac{1}{n_1}\mathbf{S_1} + \dfrac{1}{n_2}\mathbf{S}_2\)

We will reject \(H_{0}\) \colon \(\mu_{1}\) = \(\mu_{2}\) at level \(α\) if the F-value exceeds the critical value from the F-table with p and ν degrees of freedom evaluated at level \(α\).

\(F > F_{p,\nu, \alpha}\)

A reference for this particular test is given in: Seber, G.A.F. 1984. Multivariate Observations. Wiley, New York.