Maximum likelihood estimation

I really appreciate if somebody can help me to answer this question;

A sample of size n is drawn from each of two normal populations, both of which have variance ��^{2}. The mean of the two populations are (a+b) and (a-b), respectively, where a,b>0. The sample observations are denoted x_{ij, }i=1,2 and j=1,...,n. The question is " What are the maximum likelihood estimators of a,b, ��^{2}?

Thanks in advance.

Re: Maximum likelihood estimation

He Bewar.

What are the normal MLE's for the means of a distribution? Can you use the invariance principle to estimate a and b separately given their combined form?

Re: Maximum likelihood estimation

Well, could you explain it more please?

Re: Maximum likelihood estimation

The invariance principle says that if you have an estimate (say x for some parameter), then the estimation of a function of that parameter is f(x) if you use the MLE estimator.

So if you have two estimators involving a and b are both from MLE estimators, then you can create a function that will calculate the estimate for that function.

So if x = a - b and y = a + b then f(x,y) will get an MLE estimate for that function. So what functions can you create to isolate a and b if those estimations come from an MLE estimator?

Re: Maximum likelihood estimation

I see what you mean, but (a+b) and (a-b) are not going to be a function they are just means for their distribution that I have written and their distributions are normal. If you just have a look again at the question, you would see what they are. But you have more experience anyway, just solve in any way that you think. Thank you very much.