German tank problem

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 limited 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.

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.

A frequentist approach (using the minimum-variance unbiased estimator) predicts the total number of tanks produced will be: A Bayesian approach (using a uniform prior over the integers in

In many cases, statistical analysis substantially improved on conventional intelligence.

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.

[a] Shortly before D-Day, rumors indicated that large numbers of Panzer V tanks were being used.

To determine whether this was true, the Allies attempted to estimate the number of tanks being produced.

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.

[5] German records after the war showed production for the month of February 1944 was 276.

[6][c] The statistical approach proved to be far more accurate than conventional intelligence methods, and the phrase "German tank problem" became accepted as a descriptor for this type of statistical analysis.

According to conventional Allied intelligence estimates, the Germans were producing around 1,400 tanks a month between June 1940 and September 1942.

After the war, captured German production figures from the ministry of Albert Speer showed the actual number to be 245.

[3] Estimates for some specific months are given as:[7] Similar serial-number analysis was used for other military equipment during World War II, most successfully for the V-2 rocket.

[9] In the 1980s, some Americans were given access to the production line of Israel's Merkava tanks.

[11] To confound serial-number analysis, serial numbers can be excluded, or usable auxiliary information reduced.

Alternatively, sequential serial numbers can be encrypted with a simple substitution cipher, which allows easy decoding, but is also easily broken by frequency analysis: even if starting from an arbitrary point, the plaintext has a pattern (namely, numbers are in sequence).

One example is given in Ken Follett's novel Code to Zero, where the encryption of the Jupiter-C rocket serial numbers is given by: The code word here is Huntsville (with repeated letters omitted) to get a 10-letter key.

), the minimum-variance unbiased estimator (MVUE, or UMVU estimator) is given by:[e] where m is the largest serial number observed (sample maximum) and k is the number of tanks observed (sample size).

This has a variance[10] so the standard deviation is approximately N/k, the expected size of the gap between sorted observations in the sample.

The formula may be understood intuitively as the sample maximum plus the average gap between observations in the sample, the sample maximum being chosen as the initial estimator, due to being the maximum likelihood estimator,[f] with the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum,[g] and written as This can be visualized by imagining that the observations in the sample are evenly spaced throughout the range, with additional observations just outside the range at 0 and N + 1.

A derivation of the expected value and the variance of the sample maximum are shown in the page of the discrete uniform distribution.

These are easily computed, based on the observation that the probability that k observations in the sample will fall in an interval covering p of the range (0 ≤ p ≤ 1) is pk (assuming in this section that draws are with replacement, to simplify computations; if draws are without replacement, this overstates the likelihood, and intervals will be overly conservative).

The Bayesian approach to the German tank problem[14] is to consider the posterior probability

One can proceed using a proper prior over the positive integers, e.g., the Poisson or Negative Binomial distribution, where a closed formula for the posterior mean and posterior variance can be obtained.

}}d}\cdot \underbrace {{\frac {m-d}{n-d}}\cdot {\frac {m-d-1}{n-d-1}}\cdots {\frac {m-d-(k-d-1)}{n-d-(k-d-1)}}} _{k-d{\text{ times}}}={\frac {(n-k)!}{n!

The following binomial coefficient identity is used below for simplifying series relating to the German Tank Problem.

The credibility mass distribution function depends on the prior limit

is infinite The conditional probability that the largest of k observations taken from the serial numbers {1,...,n}, is equal to m, is The likelihood function of n is the same expression The total likelihood is finite for k ≥ 2: The credibility mass distribution function is The complementary cumulative distribution function is the credibility that N > x The cumulative distribution function is the credibility that N ≤ x The order of magnitude of the number of enemy tanks is The statistical uncertainty is the standard deviation