Is r^{*} linear in r?

The quantity r^{*}(\psi ; y) was introduced by Barndorff-Nielsen, in part as a distributional
refi nement of the signed likelihood root r(\psi ; y), a refi nement that can
also be approximated by a mean and standard deviation adjustment of the root
r(\psi ; y). We clarify: that r^{*} is not linear in r; that r^{*} achieves large distributional
improvement on r(\psi ; y); and that r^{*} provides the de finitive separation of inference
information concerning scalar component parameters of a statistical model.
These distributional and inference properties deserve broader awareness.