German Tank Problem - Example

Example

Suppose an intelligence officer has spotted k = 4 tanks with serial numbers, 2, 6, 7, and 14, with the maximum observed serial number, m = 14. The unknown total number of tanks is called N.

The frequentist formula for estimating the total number of tanks is

The Bayesian estimate is

\begin{align} N &\approx \mu \pm \sigma = 19.5 \pm 10 \\ \mu &= (m - 1)\frac{k - 1}{k - 2} \\ \sigma &= \sqrt{\frac{(k-1)(m-1)(m-k+1)}{(k-3)(k-2)^2}} \end{align}

(Of course we still know that they have at least 14 tanks).

The probability mass function is

\Pr(N=n) = \begin{cases} 0 &\text{if } n < m \\ \frac {k - 1}{k}\frac{\binom{m - 1}{k - 1}}{\binom n k} &\text{if } n \ge m \end{cases}

This distribution has positive skewness.

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