A discussion with British road wheel makers then estimated the number of wheels that could be produced from this many molds, which yielded the number of tanks that were being produced each month. The analysis of tank wheels yielded an estimate for the number of wheel molds that were in use. Similar analyses were done on wheels, which were observed to be sequentially numbered (i.e., 1, 2, 3, ., N). Various other components were used to cross-check the analysis. Chassis and engine numbers were also used, though their use was more complicated. The principal numbers used were gearbox numbers, as these fell in two unbroken sequences.
To do this, they used the serial numbers on captured or destroyed tanks. To determine whether this was true, the Allies attempted to estimate the number of tanks being produced. Shortly before D-Day, rumors indicated that large numbers of Panzer V tanks were being used. 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. 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. 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.
In many cases, statistical analysis substantially improved on conventional intelligence. Panther tanks are loaded for transport to frontline units, 1943ĭuring the course of the Second World 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. N m e d ≈ 74.5 Historical example of the problem The Bayesian approach predicts that the median number of tanks produced will be very similar to the frequentist prediction: The frequentist approach predicts the total number of tanks produced will be: In the SVG file, hover over a graph to highlight it.Īssuming tanks are assigned sequential serial numbers starting with 1, suppose that four tanks are captured and that they have the serial numbers: 19, 40, 42 and 60.
The example shows if four tanks are observed and the highest serial number is "60", frequentist analysis predicts 74, whereas Bayesian analysis predicts a mean of 88.5 and standard deviation of 138.72 − 88.5 = 50.22, and a minimum of 60 tanks.
Bayesian analysis has solid yellow lines with mean and shading to show range from minimum possible value to mean plus 1 standard deviation). Frequentist analysis is shown with dotted lines. The number of observations in the sample is k. Additionally, regardless of a tank's date of manufacture, history of service, or the serial number it bears, the distribution over serial numbers becoming revealed to analysis is uniform, up to the point in time when the analysis is conducted.Įstimated population size (N).
The adversary is presumed to have manufactured a series of tanks marked with consecutive whole numbers, beginning with serial number 1.
In simple terms, suppose there exists an unknown number of items which are sequentially numbered from 1 to N. In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. During World War II, production of German tanks such as the Panther was accurately estimated by Allied intelligence using statistical methods