MLE for a uniform upper bound

Question

You observe n independent samples x1, \dots, xn from a 	ext{Uniform}[0, 	heta] distribution with unknown upper endpoint 	heta. Derive the maximum likelihood estimator of 	heta, and state its bias and a corrected version.

Accepted Answer

Likelihood. Each sample contributes 1/	heta provided 0 \leq xi \leq 	heta, so L(	heta) = \begin{cases} 	heta^{-n}, & 	heta \geq \maxi xi, \ 0, & 	heta < \maxi xi. \end{cases} Maximization. Where the likelihood is nonzero, 	heta^{-n} is strictly decreasing in 	heta, so we want 	heta as small as possible. But the constraint forbids 	heta below the largest observation. The likelihood is therefore maximized at the boundary: \hat{	heta} = \maxi xi = X{(n)}. The usual "set the derivative to zero" step fails, \ell'(	heta) = -n/	heta is never zero, because the maximum sits on the edge o…

MLE for a uniform upper bound

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