Factorization theorem : Sufficient statistic - Wikipedia, the free encyclopedia
If X1, X2.......,Xn constitute a random sample of size n from an exponential population, show that "X bar" is a sufficient estimator of the parameter theta
Can somebody please, please help me with this? It is giving me a lot of trouble
Factorization theorem : Sufficient statistic - Wikipedia, the free encyclopedia
Thank you for the hint, but I already knew that factorization theorem was involved but I just can't seem to solve it. I was hoping someone could give me a more step by step answer because I am really trying but have no outside help for this course and the professor is hardly ever available to assist.
Just write your likelihood function which is the joint density....
where I left out the fact that all the x's are positive, that usually just confuses people.
meaning........
This shows that the SUM is suff for theta, but so is twice the sum....
so the sum divided by n is suff....
shows that the sample mean is suff for theta.
Not sure if I have this right but I want to make sure the way I am doing it is correct. Thanks
If X1,.......,Xn make up a random sample of size n from an exponential population, show that 'X bar' is a sufficient estimator of the parameter theta
Not sure how to write symbols, so I am going to let
Theta = Z
Summation = S
e^ = exp(...)
So,
Z^-n * exp(-S(Xi)/Z)
take LN
= -n LN(Z) - S(Xi)/Z
Take derivate w.r.t. Z
= -n/Z + S(Xi)/Z^2
Set = 0 and solve for Z
0 = -nZ + S(Xi)
nZ = S(Xi)
Z = S(Xi)/n
Is this right?