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 $(\mu_1,\mu_2,\sigma_1^2,\sigma_2^2)$?

  • $\begingroup$ Are the $X_i$s correlated wit the $Y_i$s? $\endgroup$ – Michael Chernick May 14 '18 at 20:32
  • $\begingroup$ no, they are all independent. $\endgroup$ – ChuckP May 14 '18 at 20:36
  • $\begingroup$ 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. $\endgroup$ – Michael Chernick May 14 '18 at 20:41
  • $\begingroup$ 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. $\endgroup$ – ChuckP May 14 '18 at 20:53

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