The German tank problem

MediumExpected Value Statistics~5m

An adversary stamps its tanks with distinct serial numbers $1, 2, \dots, N$ , and you do not know $N$ . Reconnaissance captures $5$ tanks with serial numbers $12,\ 47,\ 8,\ 63,$ and $90$ . Using the standard frequentist (minimum-variance unbiased) estimator, what is your best point estimate of $N$ ?

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The sample maximum $m$ is a biased-low estimate of $N$ - the true top is above everything you sampled. By how much, on average?
The $k$ observations cut $\{1,\dots,N\}$ into about $k+1$ equal gaps. Add the expected top gap: $\hat N = m(k+1)/k - 1$ .

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The German tank problem

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