I am working through find the maximum likelihood estimators of the bivariate normal distribution, without using matrices. I have the following density function:

$f(Y_1,Y_2) = \frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho_{12}^2}} \exp \bigg\{ -\frac{1}{2(1-\rho_{12}^2)} \bigg[ \bigg(\frac{Y_1 - \mu_1}{\sigma_1} \bigg)^2 -2\rho_{12} \bigg( \frac{Y_1 - \mu_1}{\sigma_1} \bigg)\bigg( \frac{Y_2 - \mu_2}{\sigma_2} \bigg) + \bigg( \frac{Y_2 - \mu_2}{\sigma_2} \bigg)^2 \bigg] \bigg\}$

So far, I've obtained the log likelihood:

\begin{equation} \begin{split} \ell(\mu_1, \mu_2, \sigma_1, \sigma_2,\rho_{12}) & = -n\log2 \pi - n\log \sigma_1 - n\log \sigma_2 - \frac{n}{2} \log(1-\rho_{12}^2) \\&- \frac{1}{2(1-\rho_{12}^2)}\sum_{i=1}^n \bigg[ \bigg(\frac{Y_1 - \mu_1}{\sigma_1} \bigg)^2 -2\rho_{12} \bigg( \frac{Y_1 - \mu_1}{\sigma_1} \bigg)\bigg( \frac{Y_2 - \mu_2}{\sigma_2} \bigg) + \bigg( \frac{Y_2 - \mu_2}{\sigma_2} \bigg)^2 \bigg] \end{split} \end{equation}

I'm having trouble with next steps: taking the partial derivatives w.r.t. each parameter, setting them equal to 0, and solving for the parameters.

Is there a better way to write the log likelihood so that I can more easily take each partial derivative?

  • 1
    $\begingroup$ Add the self study tag. $\endgroup$ Commented Feb 19, 2019 at 2:34
  • $\begingroup$ Re "is there a better way:" using matrices is helpful :-). By precluding that, you have committed yourself to performing brute-force Calculus equations. $\endgroup$
    – whuber
    Commented Feb 19, 2019 at 14:18

1 Answer 1


I guess hgupta has moved on, but for anyone else trying to work through this problem, check out Chapter 9, Section 1, Exercises 1.12-1.14 on pages 294-300 of the book An Introduction to Probability and Statistical Inference, Second Edition, written by George Roussas. He walks you through the whole problem, from deriving the estimators to verifying that they are the MLEs.

  • $\begingroup$ The book's solution is a bunch of routine calculus, which is a little bit boring. $\endgroup$
    – Tan
    Commented Jan 28, 2021 at 7:22

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