We pose a classic problem, variously known as the taxicab problem or the German tank problem (named after its historical application, by Allied forces in World War II, to the estimation of the monthly rate of German tank production from very few data).

We have a finite population $\mathcal{X}$ with labels $\{\theta +1, \theta + 2, \cdots , \theta + N\}$, where $\theta \geq 0, N \geq 1$, and both $\theta , N$ are regarded as unknown parameters. A random sample $X_1, \cdots , X_n$ is taken without replacement from $\mathcal{X}$, and suppose $X_{(1)}, X_{(n)}$ denote the minimum and the maximum of the sample labels. Let $W_n = X_{(n)}-X_{(1)}$ denote the sample range.
The problem of this issue is as follows:

a) Find in closed form an unbiased estimate $T(W_n)$ of $N$;

b) Find an unbiased estimate of the variance of $T(W_n)$;

c) Is the unbiased estimate $T(W_n)$ in part a) the UMVUE of $N$ among all possible unbiased estimates $U(X_1, \cdots , X_n)$ of $N$?

Student members of IMS are invited to submit solutions (to bulletin@imstat.org with subject “Student Puzzle Corner”). The deadline is January 29, 2020. The names of student members who submit correct solutions, and the answer, will be published in the issue following the deadline. The Puzzle Editor’s decision is final.
The solution to the previous problem is here.