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.
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.