This question is related to the below question. Estimation of logit-normal parameters
In the previous question, I showed only simple part of the whole problem.
The log-likelihood function that I would like to maximize is
$ n_1 log \int_{-\infty}^{\infty} f_1(y,\mu,\sigma) dy + n_2 log \int_{-\infty}^{\infty} f_2(y,\mu,\sigma) dy + n_3 log \int_{-\infty}^{\infty} f_3(y,\mu,\sigma) dy $
$n_1,n_2,n_3$ are known. $f_1,f_2,f_3$ are as follows.
$ f_1 = \frac{exp(2y)}{(1+exp(y))^2} \frac{1}{\sqrt{2\pi}\sigma} exp(-\frac{(y-\mu)^2}{2\sigma^2} )$
$ f_2 = \frac{exp(y)}{(1+exp(y))^2} \frac{1}{\sqrt{2\pi}\sigma} exp(-\frac{(y-\mu)^2}{2\sigma^2}) $
$ f_3 = \frac{1}{(1+exp(y))^2} \frac{1}{\sqrt{2\pi}\sigma} exp(-\frac{(y-\mu)^2}{2\sigma^2}) $
How can I find $\mu$ and $\sigma$ that maximize the above equation?
