Unsure how to find the MLE for this model The question I am trying to answer is confusing me as I don't know where to start to find a likelihood estimation
The Question
$y_i =   μ + e_i $
where the $e_i$ are independent variables distributed $N(0, σ_i^2)$
What would the maximum likelihood estimator of $μ$ be? And also, what would the mean, variance and distribution of the MLE be?
My attempt: I have tried to find the expectation of $y_i$, giving an answer of μ as the $e_i$ have mean 0. Without a distribution (i.e. Binomial, Normal) for the model, how do you find the MLE?
 A: In response to @Xi'an answer, let's assume that we have a sample size of $n=2$ with $y_1 \neq y_2$. Then
$$\sum_{i=1}^2 (y_i-\hat\mu)^{-1}=0 \Rightarrow \frac {1}{y_1-\hat \mu} + \frac {1}{y_2-\hat \mu} = 0 \Rightarrow \frac {y_2 - \hat \mu + y_1 - \hat \mu}{(y_1-\hat \mu)(y_2-\hat \mu)}= 0$$ 
$$\Rightarrow y_1 - \hat \mu + y_2 - \hat \mu = 0 \Rightarrow \hat \mu = \frac {y_1 + y_2}{2}$$
Plugging this into the derived first-order conditions for the variances, we obtain
$$\hat\sigma_1^2 = \left(y_1-\frac {y_1 + y_2}{2}\right)^2  = \frac {(y_1-y_2)^2}{4}$$
and
$$\hat\sigma_2^2 = \left(y_2-\frac {y_1 + y_2}{2}\right)^2  = \frac {(y_1-y_2)^2}{4}$$
Hmmm, so for any sample of size $n=2$, we will be estimating the two different variances as equal... Then why go into the whole trouble and treat them as different in the first place? Does this generalize in some sense for bigger $n$? Does this have any relation with the fact that we are estimating $n+1$ unknown parameters having only $n$ data points?
A: To find the maximum likelihood a generic approach is to write down the likelihood function as a starting point.
In this example, $Y_i\sim\mathcal{N}(\mu,\sigma_i^2)$ means that
$$
L(\mu,\sigma_1,\ldots,\sigma_n) = \prod_{i=1}^n \dfrac{\exp\{-(y_i-\mu)^2/2\sigma_i^2\}}{\sqrt{2\pi}\sigma_i}
$$
or
$$
\ell(\mu,\sigma_1,\ldots,\sigma_n) =\log L(\mu,\sigma_1,\ldots,\sigma_n) = \sum_{i=1}^n -(y_i-\mu)^2/2\sigma_i^2 - \log \sigma_i + \text{cst}
$$
Now that you have your (log-)likelihood function, you have to find its maximum in $(\mu,\sigma_1,\ldots,\sigma_n)$. A standard approach at this stage is to look at the derivatives against every component of the parameter:
\begin{align*}
\frac{\partial \ell}{\partial \mu}(\mu,\sigma_1,\ldots,\sigma_n)
&= \sum_{i=1}^n (y_i-\mu)/\sigma_i^2 \\
\frac{\partial \ell}{\partial \sigma_i}
(\mu,\sigma_1,\ldots,\sigma_n)
&= (y_i-\mu)^2/\sigma_i^3 - 1/\sigma_i \qquad i=1,\ldots,n
\end{align*}
If you look at the solutions $(\hat\mu,\hat\sigma_1,\ldots,\hat\sigma_n)$  of
$$
\frac{\partial \ell}{\partial \mu}(\mu,\sigma_1,\ldots,\sigma_n) =0,\qquad
\frac{\partial \ell}{\partial \sigma_i}(\mu,\sigma_1,\ldots,\sigma_n) =0,\qquad i=1,\ldots,n,
$$
you find
\begin{align*}
\sum_{i=1}^n \hat\sigma_i^{-2} y_i - n \hat\mu \sum_{i=1}^n \hat\sigma_i^{-2} &= 0\\
(y_i-\hat\mu)^2/\hat\sigma_i^3 - 1/\hat\sigma_i &= 0 \qquad i=1,\ldots,n
\end{align*}
which leads to 
$$
\hat\sigma_i^2 = (y_i-\hat\mu)^2 \qquad i=1,\ldots,n
$$
hence to
$$
\sum_{i=1}^n (y_i-\hat\mu)/\hat\sigma_i^2 = \sum_{i=1}^n (y_i-\hat\mu)^{-1}=0
$$
This equation does not have a closed form expression in $\hat\mu$.
This is the generic way to derive the likelihood equations and their solution(s). However, in some cases like the current one, the maximum of the likelihood function might occur at the boundary of the parameter space. Take indeed $\mu=y_1$. Then the log-likelihood function is equal to
$$
-\log(\sigma_1)-\sum_{i=2}^n (y_i-\mu)^2/2\sigma_i^2 -\sum_{i=2}^n \log \sigma_i + \text{cst}
$$
and if we let $\sigma_1$ go to zero, the log-likelihood goes to infinity. The same property holds for $\mu=y_2,\ldots,y_n$. The maximum of the likelihood is thus at the boundary of the parameter space
$$
\mathbb{R}\times\mathbb{R}^*_+\times\cdots\times\mathbb{R}^*_+
$$
(excluding $0$ as possible value for the $\sigma_i$) and therefore there is no maximum likelihood estimator.
