MLE at a boundary

MediumStatistics Calculus~4m

Latencies (ms) are modelled as i.i.d. Uniform $(0,\theta)$ for an unknown ceiling $\theta$ (density $1/\theta$ on $[0,\theta]$ , else $0$ ). You observe the five values $\{0.3,\,1.1,\,0.8,\,1.4,\,0.6\}$ . What is the maximum-likelihood estimate of $\theta$ ?

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$L(\theta)=\theta^{-n}$ but only for $\theta\ge\max_i x_i$ - the support constraint is everything.
$\theta^{-n}$ is decreasing, so the MLE is the smallest legal $\theta$ : the sample maximum.

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1.4 (± 0.001)

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MLE at a boundary

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