This is a self-study. I run into a distribution and I was bored so I wanted to have maximum likelihood estimators (MLE) of the parameters of this specific normal distribution but my mathematics unfortunately got me upside down.

Lets say $Y_1, Y_2,\ldots,Y_n$ iid $N(\mu, \sigma^2 \mu^2)$

I would like to estimate MLE of these parameters, especially of $\mu$.

Firstly I construct likelihood function for the distribution, which is,

$$L(\mu \mid \{Y_i\}_{i=1}^n) =\prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2\mu^2}}\exp \frac{(Y_i-\mu)^2}{\sigma^2 \mu^2}$$

The log-likelihood function is then given by,

$$\log L(\mu \mid \{Y_i\}_{i=1}^n) =\frac{-n}{2}\log{2\pi}-\dfrac{n}{2} \log{\sigma^2}-\frac{n}{2} \log{\mu^2}-\dfrac{1}{2\sigma^2\mu^2}\sum_{i=1}^2 (Y_i-\mu)^2$$

Right after then, I am all aware of the procedure that I should take partially derivatives with respect to $\mu$ and $\sigma^2$ (OR should I take derivative w.r.t $\sigma^2 \mu^2$ ???) and set the equations to zero and do some substitution and get the results. However derivatives are so complicated to solve OR there's something I am missing out. I have no idea how to get the estimators. I hope you will have an easy solution. Thanks in advance.


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