The German tank problem

Question

An enemy produces tanks numbered 1, 2, \dots, N with N unknown. You capture n tanks and read their serial numbers; the largest you see is m. What is the MLE of N, why is it biased, and how do you correct it? Work a case with n = 5 captured tanks and a maximum serial of m = 60.

Accepted Answer

The MLE. Every serial you could observe is at most N, so the likelihood is zero for any N < m and decreasing in N above m (larger N spreads the probability thinner). The maximum sits at the boundary: \hat{N}{	ext{MLE}} = m = \maxi(	ext{serials}). The bias. The largest serial in your sample almost surely falls below N, so the MLE systematically underestimates. Modeling the serials as roughly Uniform on [1, N], the expected sample maximum is about \frac{n}{n+1}N, low by a factor \frac{n}{n+1}. The correction. Scale the maximum up. The standard bias-corrected estimator adds the average gap bet…

The German tank problem

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