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The German Tank Problem assumes that we observe k i.i.d. random variables $X_1,...,X_k$ uniformly distributed over $[0,N]$, where $N$ is unknown.

The goal in this problem is to estimate $N$ given the observed data. If $m = max(X_1,...,X_k)$, then we can infer the probability mass function $$\Pr(N=n | m,k) = \begin{cases} 0 &\text{if } n < m \\ \frac {k - 1}{k}\frac{\binom{m - 1}{k - 1}}{\binom n k} &\text{if } n \ge m, \end{cases}$$ to obtain an estimate of $N$ using maximum likelihood.

Does this problem have a solution when $X_1,...,X_k$ are drawn i.i.d. from an unknown distribution with support $[0,N]$? Is there an estimate for $\hat{N}$ that we can infer even if we do not make an assumption that the data are uniform?

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Maximum-likelihood estimation involves maximisation of the likelihood function, which is not well-defined unless we have a specified class of sampling densities, parameterised (at least in part) by the parameter you want to estimate.

In the case where you merely posit that you have some distribution concentrated on $0, 1, ..., N$, this is a non-parametric problem, and you will need to be a bit clearer about exactly what class of allowable distributions you are using. You could take this to mean that you are looking at the class of all possible distributions on this discrete range with $\mathbb{P}(X=N) >0$ (i.e., the value $N$ must have positive probability, but the previous values might not be in the support). If this is the interpretation of your non-parametric class of distributions then the (trivial) maximum likelihood estimate of the distribution is the empirical distribution of the data, and the MLE is $\hat{N} = \max x_i$.

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