Empirically successful transformations from non-gaussian to close-to-gaussian distributions: Theoretical justification

Abstract

© 2016 by the Mathematical Association of Thailand. All rights reserved. A large number of efficient statistical methods have been designed for a frequent case when the distributions are normal (Gaussian). In practice, many probability distributions are not normal. In this case, Gaussian-based techniques cannot be directly applied. In many cases, however, we can apply these techniques indirectly – by first applying an appropriate transformation to the original variables, after which their distribution becomes close to normal. Empirical analysis of different transformations has shown that the most successful are the power transformations X → Xhand their modifications. In this paper, we provide a symmetry-based explanation for this empirical success.