# Minimal sufficient statistic for multivariate normal

I have the following iid. variables $X_1,..,X_n,Y_1,..,Y_m$ with distribution $X_i\sim N(\mu_1,\sigma_1^2), Y_j\sim N(\mu_2,\sigma_2^2)$. How do I find the minimal sufficient statistic for $(\mu_1,\mu_2,\sigma_1^2,\sigma_2^2)$?

• Are the $X_i$s correlated wit the $Y_i$s? – Michael Chernick May 14 '18 at 20:32
• no, they are all independent. – ChuckP May 14 '18 at 20:36
• Then wouldn't it simply be the combination of the minimal sufficient statistics for $\mu_1$ and $\sigma_1$$^2 and those for \mu_2 and \sigma_2$$^2$. Those are the respective sample means and variances. – Michael Chernick May 14 '18 at 20:41
• Is there a general way to obtain it though? because I would also like to find the minimal sufficient statistic in the case where the two mu are same, or the two sigma to be same. – ChuckP May 14 '18 at 20:53