Let \(\Theta=\mathbb{R}^{2}\), and let \(P_{\theta}\) be the von Mises-Fisher distribution on the unit circle with density

\[
P_{\theta}(d \phi)=\frac{e^{\theta^{\prime} y} d \phi}{I_{0}(|\theta|)}
\]

where \(y=(\cos \phi, \sin \phi)\) and \(d \phi\) is arc length, and \(I_{0}(\cdot)\) is the Bessel \(I\) function of order zero. Discuss the following groups as possible treatment effects acting on distributions: (i) the group of strictly positive numbers acting on \(\Theta\) by scalar multiplication; (ii) the group of planar rotations; (iii) the group of similarity transformations generated by (i) and (ii).