5 samples are drowned from uniform distribution with range [0, 2B]. The 5 samples are sorted: ($x_1 \leq x_2 \leq x_3 \leq x_4 \leq x_5$). Which of the following is an unbiased estimator for B and why?
A. $\frac{x_2+x_3}{2}$
B. $x_3$
C. $x_1$
D. $\frac{x_1+x_4}{2}$
The expectancy of each given estimator choice is the expectancy of B, but there is an explanation that since the samples are sorted, only $x_3$ is correct. Why is that? Your answers are appreciated.
It is common to use a slightly different notation for the sorted observations $$x_{(1)} \le x_{(2)} \le x_{(3)}\le x_{(4)}\le x_{(5)}$$ to distinguish them from the unsorted observations $x_1, x_2, x_3, x_4, x_5$ which were presumably sampled independently and identically from the uniform distribution. You can do the same for the associated sorted random variables $X_{(1)}, X_{(2)}, X_{(3)}, X_{(4)}, X_{(5)}$.
These sorted random variables are not independent: if $X_{(5)}$ were small the others would need to be even smaller. Nor are they identically distributed. Here we have $\frac{1}{2B} X_{(i)}\sim \text{Beta}(i,6-i)$ and $\mathbb E\left[X_{(i)}\right] = \frac i3 B$. So
A. $\mathbb E\left[\frac{X_{(2)}+ X_{(3)}}{2}\right] = \frac56B$
B. $\mathbb E\left[X_{(3)}\right] = B$
C. $\mathbb E\left[X_{(1)}\right] = \frac13B$
D. $\mathbb E\left[\frac{X_{(1)}+ X_{(4)}}{2}\right] = \frac56B$
implying that $x_{(3)}$ was the only unbiased estimate of those offered.
Alterative unbiased estimates of $B$ could have included $\frac35 x_{(5)}$, which is a function of the sufficient statistic from the sample.