The Folded Power Transformation

The folded power transformation is defined as

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

The normalized folded power transformation is defined as (Atkinson, 1985, p. 139)

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

where $G_{\mbox{\tiny f}}(\lambda )= G(y^{\lambda -1} +(1-y)^{\lambda -1})$. When you select the folded power transformation, a plot of $L(\lambda ;\bm {z}_{\mbox{\tiny f}})$ appears. You should choose a value close to the MLE value.