MLE of bivariate normal

**Sambit** · November 26th 2010, 10:16 PM

how to calculate the MLEs of the 5 parameters of a bivariate normal population, the parameters being $\mu_x,\mu_y,\sigma_x,\sigma_y,\rho$ ?

while trying to solve for $\mu_x$ , i got the following equation
$\sum{\frac{(x-\mu_x)}{\sigma_x^2}} = \rho\sum{\frac{(y-\mu_y)}{\sigma_x\sigma_y}}$

is this correct? how can i solve for $\mu_x$ from here?

**matheagle** · November 26th 2010, 10:55 PM

you need to take the partial derivative wrt to all 5 parameters
It's smart to take the logarithm of the joint density first

**Sambit** · November 27th 2010, 04:14 AM

I took the logarithm and differentiate partially wrt $\mu$ and got that

**matheagle** · November 27th 2010, 07:11 AM

you need to solve all five together, the MLE of $\mu_X$ should be $\bar X$
and those for the population variances should be the sample variances with n in the denomiator.

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