A SIMPLE MODIFICATION OF THE BOX AND COX TRANSFORMATION TO SCALE STABILITY AND INVARIANCE

We consider power-transformations to obtain stability and invariance of measurement scale. The transformation discussed here is the normalized form of the power-transformation originally suggested by Schlesselman (1971). The original suggestion is a simple modification of the Box and Cox transformation to scale invariance for measurement units. In addition to discussion on the scale invariance, we study (i) the behaviors of Jacobian of the transformation and (ii) the effect of the modification on the estimates. Then, we show that the modification of the scale invariance improves the performance of estimates, especially the mean estimate. A simulation study is per-formed to evaluate numerically performances of the modified transformation, compared with the normalized Box and Cox transformation.