I have a random variable being Fréchet distributed with shape parameter $\nu$ and scale parameter $c$ (location $0$). Now I add multiplicative noise, i.e., I set: $$ Y=X \cdot \epsilon \;, $$ with epsilon being some noise.
Now I want to calculate the maximum likelihood estimators for $\nu$ and $c$ given realizations of $Y$ and want to know in particular if the ML estimators are asymptotically consistent;
What I thought about it: If I say that $\epsilon$ is lognormally-distributed with $\mu=0$ and some $\sigma^2$, then if I take the logarithm $\log(Y)=\log(X)+\epsilon_2$ with $\epsilon_2$ being normally distributed; Now since the normal distribution with $\mu=0$ is symmetric around zero I think that the estimators of $\nu$ and $c$ might be asymptotically consistent.
Is this true? If yes - how can one see that?
Hint: Multiplicative distributions, such as the log-normal distribution you already thought of, but also the Fréchet distribution, are well characterized with geometric statistic measures.
The geometric mean of a Fréchet-distributed variable $X$ with shape $\nu$, scale $c = 1$, and location $m = 0$ is $$ \mathrm{GM}(X) = \exp\left( \dfrac{\gamma}{\nu} \right) \;, $$ where $\gamma$ is the Euler–Mascheroni constant.