An objective look at obtaining the plotting positions for QQ-plots

TL;DR

This paper objectively examines methods for selecting plotting positions in QQ-plots across three frameworks, clarifying multiple legitimate solutions and providing guidance for practitioners based on their objectives.

Contribution

It introduces a comprehensive analysis of plotting positions in QQ-plots using distribution-theoretic, decision-theoretic, and game-theoretic frameworks, highlighting multiple valid solutions.

Findings

Multiple legitimate solutions depend on the practitioner's objective.

The paper derives plotting positions within three different frameworks.

It discusses approximations when closed-form solutions are unavailable.

Abstract

Choosing the plotting positions for the QQ-plot has been a subject of much debate in the statistical and engineering literature. This paper looks at this problem objectively by considering three frameworks: distribution-theoretic; decision-theoretic; game-theoretic. In each framework, we derive the plotting positions and show that there are more than one legitimate solution depending on the practitioner's objective. This work clarifies the choice of the plotting positions by allowing one to easily find the mathematical equivalent of their view and choose the corresponding solution. This work also discusses approximations to the plotting positions when no closed form is available.