Finding MLE of the common $\mu$ from normal samples with two unknown variances

by moreblue   Last Updated October 09, 2018 11:19 AM

My problem is as follows,

Find the maximum likelihood estimator, $$\mu^{MLE}$$ from $$(m+n)$$ samples, where $$X_1, \cdots, X_m \sim N(\mu, \sigma_1^2), Y_1, \cdots, Y_n \sim N(\mu, \sigma_2^2),$$

where $$\sigma_1^2, \sigma_2^2$$ : both unknown.

My attempt

\begin{aligned} \log L(\mu) = l(\mu) &= \mathrm{constant}-\frac{1}{2} \left[\sum_{i=1}^m\frac{(x_i - \mu)^2}{\sigma_1^2} + \sum_{j=1}^n\frac{(y_j - \mu)^2}{\sigma_2^2} \right] \\ \frac{\partial l(\mu)}{\partial \mu} &= - \frac{\mu}{\sigma_1^2} \sum_{i=1}^m x_i^2 - \frac{\mu}{\sigma_2^2} \sum_{j=1}^n y_j^2 + \frac{m \bar{x}}{\sigma_1^2} + \frac{n \bar{y}}{\sigma_2^2} \end{aligned}

This yields $$\hat{\mu^{MLE}} =\frac{ m \bar{x}/\sigma_1^2 +n\bar{y}/\sigma_2^2 }{m /\sigma_1^2 +n/\sigma_2^2}$$, but the main problem is, I basically don't know the variances. Any help will be appreciated.

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