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Need help with estimators (Statistics)

We have that X1 to Xn are i.i.d Uniform (0, theta) random variables, where theta is greater than zero but unknown. The question is to explain why (2/n)(Sum from i=1 to i=n of Xi) is not a sufficient statistic. Any ideas? I have no clue to be honest.

Sorry for the poor notation, hope it's legible.
Original post by pineapplechemist
We have that X1 to Xn are i.i.d Uniform (0, theta) random variables, where theta is greater than zero but unknown. The question is to explain why (2/n)(Sum from i=1 to i=n of Xi) is not a sufficient statistic. Any ideas? I have no clue to be honest.

Sorry for the poor notation, hope it's legible.


I'll try and get you started! So,

(i) What is the definition of sufficiency and what does it mean?

(ii) Can you think of a big theorem that might help us here?
Original post by Gregorius
I'll try and get you started! So,

(i) What is the definition of sufficiency and what does it mean?

(ii) Can you think of a big theorem that might help us here?


(i) It's to do the the conditional distribution depending on the unknown parameter or not

(ii) I'm not entirely certain: I was thinking along the lines of the factorization theorem but I'm not 100 percent sure. I think the only other theorem I'm aware of to do with sufficiency is Rao-Blackwell: would this help?
Original post by pineapplechemist
(i) It's to do the the conditional distribution depending on the unknown parameter or not

(ii) I'm not entirely certain: I was thinking along the lines of the factorization theorem but I'm not 100 percent sure. I think the only other theorem I'm aware of to do with sufficiency is Rao-Blackwell: would this help?


You're getting close. A statistic is sufficient for a parameter θ\theta if the conditional distribution of the sample given the value of the statistic is independent of θ\theta. What this means informally is that the statistic contains all the information about θ\theta that is present in the sample.

The factorization theorem is a key tool here (not Rao-Blackwell, as that is about improving estimators by conditioning on a sufficient statistic). It's key because it's an "if and only if" theorem.

So, can you write down the joint distribution of the sample given θ\theta? Then have a good think and see if it can factor in the way that the theorem requires.
Original post by Gregorius
You're getting close. A statistic is sufficient for a parameter θ\theta if the conditional distribution of the sample given the value of the statistic is independent of θ\theta. What this means informally is that the statistic contains all the information about θ\theta that is present in the sample.

The factorization theorem is a key tool here (not Rao-Blackwell, as that is about improving estimators by conditioning on a sufficient statistic). It's key because it's an "if and only if" theorem.

So, can you write down the joint distribution of the sample given θ\theta? Then have a good think and see if it can factor in the way that the theorem requires.


Well the joint distribution given theta is 1/(theta)^n right? (by independence). I'm still unsure as to what to do with my T(X) though.
Original post by pineapplechemist
Well the joint distribution given theta is 1/(theta)^n right? (by independence). I'm still unsure as to what to do with my T(X) though.


You have nearly got the joint distribution. But notice that it is non zero on a restricted domain that you can write in terms of the xix_i. That is, you need an indicator function in there.

Apologies in advance but I am going to be away from computers for much of the rest of the day, so won't be able to help again (if you need it) until tomorrow. A google search on sufficient statistic uniform should give you some useful stuff if you are still stuck!
Original post by Gregorius
You have nearly got the joint distribution. But notice that it is non zero on a restricted domain that you can write in terms of the xix_i. That is, you need an indicator function in there.

Apologies in advance but I am going to be away from computers for much of the rest of the day, so won't be able to help again (if you need it) until tomorrow. A google search on sufficient statistic uniform should give you some useful stuff if you are still stuck!


Thanks for your help!

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