1. Which estimator is better?

Suppose that n=5 observations are taken from the uniform pdf, f_Y(y;theta)=1/theta, 0<=y<=theta, where theta is unknown. Two unbiased estimators for theta are theta hat_1 = (6/5)*Y_max and
theta hat_2 = 6*Y_min. Which estimator would be better to use? Does the answer as to which estimator is better make sense on intuitive grounds? Explain.

Hint: What must be true of Var(Y_max) and Var(Y_min) given that f_Y(y;theta) is symmetric? Note that Var(Y_max) and Var(Y_min) do not need to be formally calculated.

2. Originally Posted by eigenvector11
Suppose that n=5 observations are taken from the uniform pdf, f_Y(y;theta)=1/theta, 0<=y<=theta, where theta is unknown. Two unbiased estimators for theta are theta hat_1 = (6/5)*Y_max and
theta hat_2 = 6*Y_min. Which estimator would be better to use? Does the answer as to which estimator is better make sense on intuitive grounds? Explain.

Hint: What must be true of Var(Y_max) and Var(Y_min) given that f_Y(y;theta) is symmetric? Note that Var(Y_max) and Var(Y_min) do not need to be formally calculated.
You should be learning enough by now from the replies to all your other questions that you can show some working here and say where you get stuck.

3. Well the thing is, I don't think the question wants us to show any work, since it says that we don't have to formally calculate the Variances. I know that one estimator is more efficient than the other if it's variance is less. But without actually finding the variances of the estimators, I am unsure of how to answer this question. Intuitively, I would say that 6*Y_min would be more efficient, since taking the least value of a set of numbers and multiplying by 6 would be less than taking the largest and multiplying by (6/5). I would think that my prof is looking for more of a sophisticated answer than this though.

4. Originally Posted by eigenvector11
Well the thing is, I don't think the question wants us to show any work, since it says that we don't have to formally calculate the Variances. I know that one estimator is more efficient than the other if it's variance is less. But without actually finding the variances of the estimators, I am unsure of how to answer this question. Intuitively, I would say that 6*Y_min would be more efficient, since taking the least value of a set of numbers and multiplying by 6 would be less than taking the largest and multiplying by (6/5). I would think that my prof is looking for more of a sophisticated answer than this though.
Since my statistics is not great, I would just do the calculation of the variance for each and hope that I got some insight in hindsight.