Let $(X_1, Y_1), . . . ,(X_n, Y_n)$ be a random sample from a bivariate normal distribution
$N(\begin{bmatrix}0\\0\end{bmatrix} , \begin{bmatrix}\sigma^2&p\sigma^2\\p\sigma^2&\sigma^2\end{bmatrix}) $
where $\sigma^2 >0$ and $p \in (−1, 1)$ are unknown parameters. Find the MLE’s of $\sigma^2 >0$ and $p$
So I tried doing this the normal way, finding $\prod_{i=1}^n f(x,y).$ then finding the derivatives of both $p$ and $\sigma^2$, however that got me nowhere. Is there any other method of soliving this problem?
Since you are assuming that both $X$ and $Y$ follows a normal distribution with mean $0$ and variance $\sigma^2$ with a correlation of $p\sigma^2$ between the two random variables, the maximum likelihood estimate of $p$ should be $p_{o} = \frac{1}{\sigma^2_{o}} \frac{\frac{1}{n}\sum\limits_{i=1}^{n} x_i y_i}{\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n} x_i^2}\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n} y_i^2}} = \frac{\rho}{\sigma^2_{o}}$ and maximum likelihood of $\sigma^{2}$ should be $\sigma^2_{o} = \frac{1}{n} \sum\limits_{i=1}^{n} (x_i^2 + y_i^2) + 2p_{o}\sigma^{2}_{o} = \frac{1}{n} \sum\limits_{i=1}^{n} (x_i^2 + y_i^2) + \rho$. So the putting MLE of $\sigma^{2}$ in the expression of $p_{o}$, we can get an expression of MLE of $p$.
I think this should be the result of taking derivative of the log likelihood expression with respect to $p$ and $\sigma^2$ and equating the resulting equations to zero and solving them.
Note that I used the formulas of correlation between two random variables and variance of sum of random variables. Let me know if you want me to write down the formulas and derive it from log likelihood expression (it'll take some time).