$n$ is an integer greater than 2.
$\mu\neq 0,\gamma \gt 0$ are real number.
$X_1,…,X_n$ are random variables and each $X_i (i=1,…,n)$ follows normal distribution with mean $\mu$, and variance $\frac{\gamma}{\mu^2}$.
What are MLE of $\mu$, $\hat{\mu}$?
What I have tried
$$f(x)=\frac{1}{\sqrt{2\pi}\frac{\gamma}{\mu^2}}\exp{-\frac{ (x_i-\mu)^2}{2\frac{\gamma}{\mu^2}}}\\\ L=\frac{1}{(2\pi)^{\frac{n}{2}}(\frac{\gamma}{\mu^2})^n}\exp{-\frac{\displaystyle \sum_{i=1}^n (x_i-\mu)^2}{2\frac{\gamma}{\mu^2}}}\\\ l=-\frac{n}{2}\log{2\pi}-n\log{\frac{\gamma}{\mu^2}}-\frac{\displaystyle \sum_{i=1}^n (x_i-\mu)^2\mu^2}{2\gamma}\\\ \frac{\partial l}{\partial \mu}=0$$
I could not solve the last equation.
Best Answer
Since $\partial_\mu(x_i\mu-\mu^2)=x_i-2\mu$, $\partial_\mu(x_i\mu-\mu^2)^2=2\mu(x_i-\mu)(x_i-2\mu)$, so$$\partial_\mu l=\frac{2n}{\mu}-\frac{\mu}{\gamma}\sum_i(x_i-\mu)(x_i-2\mu).$$So $\partial_\mu l=0$ reduces to solving a quartic equation in $\mu$.