Hi there!
This page is about a transformation that may be applied to a random variable
which has a positively or negatively skewed distribution. The transformation may
be useful for normalizing this distribution, at least in terms of it's third moment.

Historically, Box and Cox suggested the following transformation of a random variable,
which we will call eta.
T(eta)=(eta^lambda-1)/lambda ; eta>0 and lambda <>0
Note that this transform has several annoying features,

  1. Not defined for when eta=0.
  2. Not differentiable with respect to eta about eta=zero.
  3. And one nice one, basically as lambda approaches zero from above we get the log transform.

Ideal properties of such a transform are...

  1. Invertible.
  2. Defined for all eta.
  3. Does not produce imaginary components as long as eta has none.
  4. One that can deal with positive or negative skewness.
  5. One that is differentiable everywhere.

Here is a suggestion, which by Stigler's Law of Eponomy has an unknown creator
T(eta)=(exp^(lambda*eta)-1)/lambda Properties 1,2,3, and 5 can be easily proven. Property 4 is for you to experiment with!