  Some 40 years ago Khmaladze introduced a transform which greatly facilitated
the distribution free goodness of fit testing of statistical hypotheses. In the
last decade, he has published a related transform, broadly offering an
alternative means to the same end. The aim of this paper is to derive these
transforms using relatively elementary means, making some simplifications, but
losing little in the way of generality. In this way it is hoped to make these
transforms more accessible and more widely used in statistical practice. We
also propose a change of name of the second transform to the Khmaladze
rotation, in order to better reflect its nature.
