Two unbiased estimators, which one is efficient?

Question

Sample X1, \dots, Xn i.i.d. from 	ext{Uniform}(0, 	heta). Two estimators of 	heta are both unbiased: the moment estimator \hat	heta1 = 2\bar X, and the rescaled maximum \hat	heta2 = \frac{n+1}{n} X{(n)} where X{(n)} = \maxi Xi. Verify both are unbiased, then decide which to use and why.

Accepted Answer

Both are unbiased. E[X] = 	heta/2 so E[2\bar X] = 	heta. And E[X{(n)}] = \frac{n}{n+1}	heta, so E\!\left[\frac{n+1}{n}X{(n)}ight] = 	heta. Now the variances. \operatorname{Var}(\hat	heta1) = 4\operatorname{Var}(\bar X) = 4\cdot\frac{	heta^2/12}{n} = \frac{	heta^2}{3n}, which shrinks like 1/n. For the maximum, \operatorname{Var}(X{(n)}) = \frac{n	heta^2}{(n+1)^2(n+2)}, so \operatorname{Var}(\hat	heta2) = \left(\frac{n+1}{n}ight)^2 \frac{n	heta^2}{(n+1)^2(n+2)} = \frac{	heta^2}{n(n+2)}, which shrinks like 1/n^2. For any n 1 the max-based estimator has far smaller variance (for n …

Two unbiased estimators, which one is efficient?

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