A Transformation Characterizing the Normal Distribution

Abstract

In this article we propose a new transformation of random variables (RVs) which characterizes the normal distribution. It allows us to transform n i.i.d. normal RVs whose mean and variance are unknown into new n − 2 i.i.d. new normal variables with zero mean while maintaining the same unknown variance. This belongs to the class of transformations designed to reduce the number of unknown parameters or remove them altogether.

Some historical remarks concerning methods for removing parameters in the normal distribution are given and two possible applications of the new transformation are described.

Keywords:

• Characterization of normality
• Random variable transformations
• Removal of nuisance parameters
• Testing for normality

Mathematics Subject Classification:

• Primary 62
• Secondary E, 15, 10

Acknowledgments

I would like to express my sincere acknowledgements to three reviewers for comments and suggestions which have helped me improve this presentation.