If I have a Gamma distributed random variable $y$:
\begin{equation} y \sim Gamma(\alpha, \beta) \end{equation}
which has coefficient of variation $\frac{1}{\sqrt{\alpha}}$, then what would be the coefficient of variation of $x$, defined as:
\begin{equation} x = \ln(y) \end{equation}
? I have found some information on the log-Gamma distribution, but it isn't clear to me whether the coefficient of variation is just $\ln( \frac{1}{\sqrt{\alpha}})$.
Thanks!
The logarithm of $Y$ has momentgenerating function \begin{align} M(t) &= Ee^{t\ln Y} \\ &= E(Y^t) \\ &= \int_0^\infty \frac{\lambda^\alpha}{\Gamma(\alpha)}y^{\alpha + t - 1}e^{-\lambda t}dt \\ &= \frac{\lambda^\alpha\Gamma(\alpha+t)}{\Gamma(\alpha)\lambda^{\alpha+t}} \\ &= \frac{\Gamma(\alpha+t)}{\Gamma(\alpha)\lambda^t} \end{align} and cumulant generating function \begin{align} K(t)=\ln M(t)=\ln\Gamma(\alpha + t)-\ln\Gamma(\alpha)-t\ln\lambda. \end{align} Hence, \begin{align} E(\ln Y)&=K'(0)=\psi_0(\alpha)-\ln\lambda, \\ \operatorname{Var}(\ln Y)&=K''(0)=\psi_1(\alpha), \end{align} and $$ \operatorname{CV}(\ln Y)=\frac{\sqrt{\psi_1(\alpha)}}{\psi_0(\alpha)-\ln\lambda}, $$ where $\psi_0$ and $\psi_1$ are the di- and trigamma functions.
