Minimal sufficient statistics for a two sample variable with normal distribution

Consider is a random sample from and is an independent random sample from

Let denote the joint distribution of these n+m variables with . Find a minimal sufficient statistics for this family of distributions.

My solution so far:

Now, we know that the joint density function for X and Y is:

First, since the Xi's are not independent, I'm not even sure if I can do this. Second, now I can't really compass my statistic , mainly to get the Xi's and Yj's together. Any hints? Thank you very much!

Re: Minimal sufficient statistics for a two sample variable with normal distribution

I would think that all n+m random variables are independent.

I would expect to see that the sample variances are

while the mean is a pooled estimator

Re: Minimal sufficient statistics for a two sample variable with normal distribution

Thanks for your help, but I'm just having trouble in attempt to mold those terms into what you suggested the estimator is.