  We examine the normal approximation of the modified likelihood root, an
inferential tool from higher-order asymptotic theory, for the linear
exponential and location-scale family. We show that the $r^\star$ statistic can
be thought of as a location and scale adjustment to the likelihood root $r$ up
to $O_p(n^{-3/2})$, and more generally $r^\star$ can be expressed as a
polynomial in $r$. We also show the linearity of the modified likelihood root
in the likelihood root for these two families.
