On the derivation of the Khmaladze transforms

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.