Variable Transformations

The Guerrero-Johnson Transformation

The Guerrero-Johnson transformation is defined as

\[ \mbox{GJ}(y;\lambda ) = \left\{ \begin{array}{l l} \frac{p^\lambda - 1}{\lambda } & \mbox{if } \lambda \neq 0 \\ \log (p) & \mbox{if } \lambda = 0 \end{array} \right. \]

The normalized Guerrero-Johnson transformation is defined as (Atkinson 1985, p. 145)

\[ \bm {z}_{\scriptscriptstyle GJ}(\lambda ; y) = \left\{ \begin{array}{l l} \frac{p^{\lambda } - 1}{\lambda G_{\scriptscriptstyle GJ}(\lambda )} & \mbox{if } \lambda \neq 0 \\ \log (p) G(y(1-y)) & \mbox{if } \lambda = 0 \end{array} \right. \]

where $G_{\scriptscriptstyle GJ}(\lambda )= G(y^{\lambda -1}/(1-y)^{\lambda +1})$. When you select the Guerrero-Johnson transformation, a plot of $L(\lambda ;\bm {z}_{\scriptscriptstyle GJ})$ appears. You should choose a value close to the MLE value.