I have a question on this famous problem called the German tank problem.
Assuming tanks are assigned sequential serial numbers starting with 1, suppose that four tanks are captured and that they have the serial numbers: 19, 40, 42 and 60. What's your best guess for $N$, the the total number of tanks that are produced?
In this article, deriving the frequentist approach for this problem, the author starts by finding the expected value of not $N$, but the sample max in terms of $N$. Then, he rearranges the formula in terms of $N$, and that's the estimate. According to the Wikipedia, this method not only gives an unbiased estimator, but the minimum-variance unbiased estimator. Why?