🔗 The German tank problem

🔗 Mathematics 🔗 Germany 🔗 Military history 🔗 Statistics 🔗 Military history/Intelligence 🔗 Military history/World War II 🔗 Military history/German military history 🔗 Military history/European military history

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 we have an unknown number of items which are sequentially numbered from 1 to N. We take a random sample of these items and observe their sequence numbers; 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 the estimation of the monthly rate of German tank production from very few data. This exploited the manufacturing practice of assigning and attaching ascending sequences of serial numbers to tank components (chassis, gearbox, engine, wheels), with some of the tanks eventually being captured in battle by Allied forces.