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

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.

KEYWORDS:

• Distribution-free
• Invariant
• Loss function
• Plotting position
• QQ-plot.

Mathematics Subject Classification:

• 62G30, 62-09

Acknowledgments

We would like to thank Prof. Jim Zidek, Prof. Nhu Le and Prof. David Scott for suggestions which have improved this work. The first author was partially supported by research grants from Japanese Society for Promotion of Science. We would also like to thank the anonymous referees for reviewing the manuscript carefully and making valuable suggestions which have resulted an improvement of this paper.