For linear models $$y_{n \times1}=X_{n \times p}\beta_{p \times 1}+\epsilon_{n \times 1}, \text{ where }\epsilon \sim N(0,V)$$ If in a real life problem we have data as $(y_1,x_1),(y_2,x_2),...,(y_n,x_n)$ where $x_i$'s are row vectors in $X$, and we want to fit the data using the linear model above. Then how could we get a estimate of $V$?
I know usually, we are interested in estimating $\beta$ and the covariance of $\beta$. But here I want to find out the distribution of $y$, say $y \sim N_n(X\beta, V)$, then what is $V$? Is there an estimate of it?
For me, the vector $y$ is just one observation from the multinormal distribution $y \sim N_n(X\beta, V)$. Like in the univariate normal case, if we have an observation $z$ from $N(\mu, \sigma^2)$, we can estimate $\mu$ as $\hat{\mu}=\bar{z}=z$, but we can not estimate $\sigma^2$ since we only have one observation. So does that mean we cannot estimate $V$ here?
Moreover, if we have a linear mixed effect model $$y=X\beta+Z\alpha+\epsilon$$ where $\beta$ is the fixed effect and $\alpha$ is the random effects, and $$\epsilon \sim N(0, V) \text{ and }\alpha \sim N(0, \Omega)$$ Then is there a way to estimate $V$ and $\Omega$? Then we could find the covariance matrix of $y$.
Thanks!