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Thread: MLE of bivariate normal

  1. #1
    Senior Member Sambit's Avatar
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    Question MLE of bivariate normal

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

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

    is this correct? how can i solve for $\displaystyle \mu_x$ from here?
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  2. #2
    MHF Contributor matheagle's Avatar
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    you need to take the partial derivative wrt to all 5 parameters
    It's smart to take the logarithm of the joint density first
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  3. #3
    Senior Member Sambit's Avatar
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    I took the logarithm and differentiate partially wrt $\displaystyle \mu$ and got that
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  4. #4
    MHF Contributor matheagle's Avatar
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    you need to solve all five together, the MLE of $\displaystyle \mu_X$ should be $\displaystyle \bar X$
    and those for the population variances should be the sample variances with n in the denomiator.
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