Inverse Normalizing Transformations and An Extended Normalizing Transformation

When available data are non-normal, a common practice is to normalize them by applying the Box-Cox power transformation. The general effectiveness of this transformation implies that an inverse normalizing transformation, viz. a power transformation of the standard normal quantile, may effectively deliver a general representation for many of the commonly applied theoretical statistical distributions.
Employing as a departure point the Box-Cox transformation, we develop in this paper several inverse normalizing transformations (INTs), and define criteria for their effectiveness. In terms of these criteria, the new transformations are shown to deliver good representations to differently shaped distributions having skewness values that range from zero to over 11. A new normalizing transformation, derived by conversion of a certain INT, turns out to be an extension of the classical Box-Cox transformation. The new normalizing transformation provides an appreciably better normalizing effect relative to the Box-Cox transformation. Some estimation procedures for the new INTs are developed.