German tank problem

In the statistical theory of estimation, the German tank problem involves estimating the maximum of a discrete uniform distribution from sampling without replacement. It is named from its application in World War II to the estimation of the number of German tanks.

This analysis shows the approach that was used and illustrates the difference between frequentist inference and Bayesian inference.

Estimating the population maximum based on a single sample yields divergent results, while the estimation based on multiple samples is an instructive practical estimation question whose answer is simple but not obvious.

Suppose k = 4 tanks with serial numbers 19, 40, 42 and 60 are captured. The maximal observed serial number, m = 60. The unknown total number of tanks is called N.

The formula for estimating the total number of tanks suggested by the frequentist approach outlined below is

whereas the Bayesian analysis below yields (primarily) a probability mass function for the number of tanks

from which we can estimate the number of tanks according to

This distribution has positive skewness, related to the fact that there are at least 60 tanks.

During the course of the war, the Western Allies made sustained efforts to determine the extent of German production and approached this in two major ways: conventional intelligence gathering and statistical estimation. In many cases, statistical analysis substantially improved on conventional intelligence. In some cases, conventional intelligence was used in conjunction with statistical methods, as was the case in estimation of Panther tank production just prior to D-Day.

The allied command structure had thought the Panzer V (Panther) tanks seen in Italy, with their high velocity, long-barreled 75 mm/L70 guns, were unusual heavy tanks and would only be seen in northern France in small numbers, much the same way as the Tiger I was seen in Tunisia. The US Army was confident that the Sherman tank would continue to perform well, as it had versus the Panzer III and Panzer IV tanks in North Africa and Sicily. Shortly before D-Day, rumors indicated that large numbers of Panzer V tanks were being used.

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