Variable Transformations

The Aranda-Ordaz Transformation

The Aranda-Ordaz transformation is defined as

\[ \mbox{AO}(y;\lambda ) = \left\{ \begin{array}{l l} \frac{2 (p^\lambda - 1)}{\lambda (p^\lambda +1)} & \mbox{if } \lambda \neq 0 \\ \log (p) & \mbox{if } \lambda = 0 \end{array} \right. \]

The normalized Aranda-Ordaz transformation is defined as (Atkinson 1985, p. 149)

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

where $G_{\scriptscriptstyle AO}(\lambda )= G(2p^{\lambda -1}(1+p)^2/(p^\lambda +1)^2)$. When you select the Aranda-Ordaz transformation, a plot of $L(\lambda ;\bm {z}_{\scriptscriptstyle AO})$ appears. You should choose a value close to the MLE value.