Maximum likelihood estimation

Oct 2012
5
0
The UK
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 xij, i=1,2 and j=1,...,n. The question is " What are the maximum likelihood estimators of a,b, 2?

Thanks in advance.



















 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
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?
 
Oct 2012
5
0
The UK
Well, could you explain it more please?
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
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?
 
Oct 2012
5
0
The UK
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.