Arbitrary Scaling

As we mentioned, the mean of vector ${\bf x}$ is assumed to be equal to the degrees of freedom ${\bf k}$. If the reference hypothesis $H_0$ assumes a different mean value, then we can regard the feature transformation as having been applied to a scaled variable. This is easily handled by the change of values theorem. Let ${\bf x}$ have the prescribed degrees of freedom, but mean $\mu$. Then, we write that

$${\bf z}= {\bf A}^\prime {\bf x}= {\bf A}^\prime$$   $\Lambda$$$\tilde{{\bf x}},$$

where $\tilde{{\bf x}}$ has the standard distribution with mean ${\bf k}$, and $\Lambda$ is the diagonal matrix with elements $\Lambda$$_i = \mu_i/k_i.$ Then, we simply replace matrix ${\bf A}$ with $\Lambda$${\bf A}$ and solve the problem assuming ${\bf x}$ has the standard distribution with mean ${\bf k}$.