On the derivation of the Khmaladze transforms

TL;DR

This paper derives and simplifies Khmaladze transforms, making them more accessible for statistical goodness-of-fit testing, and proposes renaming the second transform as the Khmaladze rotation.

Contribution

It provides elementary derivations and simplifications of Khmaladze transforms, enhancing their accessibility and practical use in statistics.

Findings

Simplified derivations of Khmaladze transforms

Proposed renaming to Khmaladze rotation

Enhanced accessibility for statistical practice

Abstract

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.