MLE for a uniform's upper endpoint

Question

You observe X1, \dots, Xn i.i.d. from 	ext{Uniform}(0, 	heta) with 	heta unknown. Derive the MLE of 	heta. Is it biased, and if so, by how much?

Accepted Answer

Likelihood. Each observation has density \frac{1}{	heta} on [0, 	heta] and 0 outside, so L(	heta) = \prod{i=1}^n \frac{1}{	heta}\,\mathbf{1}\{0 \le Xi \le 	heta\} = 	heta^{-n}\,\mathbf{1}\{	heta \ge \maxi Xi\}. Maximize (no calculus). The factor 	heta^{-n} is strictly decreasing in 	heta, so you want 	heta as small as possible, but the indicator forbids 	heta < \maxi Xi (that would assign zero probability to data you actually observed). So the likelihood is maximized at the boundary: \hat	heta = \maxi Xi = X{(n)}. This is why the derivative-and-set-to-zero recipe fails, the maximu…

MLE for a uniform's upper endpoint

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