Solution
MLE for the Uniform
Show the problem again
You observe n independent draws from a Uniform(0, θ) distribution with θ unknown. What is the maximum likelihood estimator of θ?
Worked solution
The MLE is the sample maximum, θ̂ = max(X₁, ..., Xₙ), and it is biased low. The likelihood is θ⁻ⁿ whenever θ ≥ all observations and 0 otherwise, so it is maximized by taking θ as small as legally possible: the sample maximum. Since the max is almost surely strictly below θ, E[θ̂] = nθ/(n+1) < θ. Multiplying by (n+1)/n fixes the bias, which is precisely the German tank logic in continuous form.
Source: Standard maximum-likelihood exercise from mathematical statistics texts. Statement written for AxiomIQ.