1. ## Sufficient estimator

Consider the uniform distribution on $\displaystyle [0,\theta]$

Define $\displaystyle G_n=\sqrt[n]{Y_1\cdots Y_n}$ (i.e. the geometric mean) and the Ys are independent rvs.

Is $\displaystyle G_n$ a sufficient estimator?

I just really have a hard time with the concept of sufficiency. I basically just try to use the factorization theorem without understanding what I'm doing, and I am struggling to do so with the geometric mean. I was thinking to use an indicator function for y from 0 to theta?

EDIT: Maybe I am having trouble because it is not sufficient. It seems like averages are not going to tell all the information relevant to find upper bounds.

2. ## Sufficient estimator - index functions

It looks like we are covering the same thing in our class right now...

It looks right to use the indicator function, but you won't be able to use the Factorization Theorem with an indicator with respect to y that depends on theta (doesn't separate out nicely).

I have found that sometimes you can just keep going with the indicator idea and it reduces down by itself or you can use the definition for minimal sufficient statistic to get you on the way.

I'll need to leave the actual answer to someone else - I'm still too slow at figuring all this out Hope that helps though.

3. Thanks for the advice...I just don't think G is sufficient. For example if you take a sample of size 2 and you get 12 and 4, then G estimates theta as $\displaystyle \sqrt{48}$ If you take another sample of size 2 and get 48 and 1, G gives the same estimate for theta. However, the second sample clearly gives you more information on what theta can be.

4. The largest order stat is suff for theta.