The problem of estimating the maximum N of a discrete uniform distribution on the integer interval [1,N] from a sample of
k observations is commonly known as the
German tank problem, following the practical application of this maximum estimation problem, during
World War II, by Allied forces seeking to estimate German tank production. A
uniformly minimum variance unbiased (UMVU) estimator for the distribution's maximum in terms of
m, the
sample maximum, and
k, the
sample size, is \hat{N}=\frac{k+1}{k} m - 1 = m + \frac{m}{k} - 1. This can be seen as a very simple case of
maximum spacing estimation. This has a variance of \frac{1}{k}\frac{(N-k)(N+1)}{(k+2)} \approx \frac{N^2}{k^2} \text{ for small samples } k \ll N so a standard deviation of approximately \tfrac N k, the population-average gap size between samples. The sample maximum m itself is the
maximum likelihood estimator for the population maximum, but it is biased. If samples from a discrete uniform distribution are not numbered in order but are recognizable or markable, one can instead estimate population size via a
mark and recapture method. ==Random permutation==