I have this mixture distribution
$f(x) =w \cdot \mathcal{LN}(\mu_1,\sigma) + (1-w)\cdot \mathcal{LN}(\mu_2,\sigma) $
where $\mathcal{LN}(\mu,\sigma)$ is a lognormal distribution.
I now have random variables
$ Y = \sum_{i=0}^n X_i $ where all $X_i$ are distributed according to $f(x)$.
How can I get the maximum likelihood estimation for the unknown paramters $\mu_1, \mu_2$ and $\sigma$ (w is given)? We can also assume that $n$ is a small number, for example $n=10$.
Are there other methods of finding the parameters?