German tank problem

In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. In simple terms, suppose there exists an unknown number of items which are sequentially numbered from 1 to N. A random sample of these items is taken and their sequence numbers observed; the problem is to estimate N from these observed numbers. The problem can be approached using either frequentist inference or Bayesian inference, leading to different results. Estimating the population maximum based on a single sample yields divergent results, whereas estimation based on multiple samples is a practical estimation question whose answer is simple (especially in the frequentist setting) but not obvious (especially in the Bayesian setting). The problem is named after its historical application by Allied forces in World War II to estimate the monthly rate of German tank production from very limited data. The statistical method exploited the manufacturing practice of assigning and attaching ascending serial-number sequences to tank components (chassis, gearbox, engine, wheels), with some of the tanks eventually captured in battle by Allied forces. This contrasted with conventional intelligence, which often overestimated Nazi output; this statistical approach yielded a highly accurate estimate.