Let \(Y\) be a non-negative random variable with cumulants \(\kappa_{r}\) such that \(\kappa_{r} / \kappa_{1}^{r}=\) \(O\left(ho^{r-1}ight)\) as \(ho ightarrow 0\). In other words, the scale-free variable \(Z=Y / \kappa_{1}\) has variance \(ho=\kappa_{2} / \kappa_{1}^{2}\), which is the squared coefficient of variation of \(Y\), and the higher-order scale-free cumulants are \(O\left(ho^{r-1}ight)\). Show that the cumulants of the power-transformed variable are

\[
\begin{aligned}
E\left(Z^{\lambda}ight) & =1+\frac{(\lambda-1) \kappa_{2}}{2 \kappa_{1}^{2}}+o(ho) \\
\operatorname{var}\left(Z^{\lambda}ight) & =\frac{\kappa_{2}}{\kappa_{1}^{2}}+o(ho) \\
\operatorname{cum}_{3}\left(Z^{\lambda}ight) & =\frac{\kappa_{3}}{\kappa_{1}^{3}}+3(\lambda-1) \frac{\kappa_{2}^{2}}{\kappa_{1}^{2}}+o\left(ho^{2}ight) .
\end{aligned}
\]

Hence deduce that the approximate symmetry-inducing power transformation is \(\hat{\lambda}=\) \(1-\kappa_{1} \kappa_{3} /\left(3 \kappa_{2}^{2}ight)\).