The biased MLE of Uniform(0, θ)

Question

You observe X1, \dots, Xn i.i.d. Uniform(0, 	heta) with 	heta unknown. Find the MLE of 	heta, show it is biased, correct the bias, and compare the corrected estimator to the method-of-moments alternative 2\bar{X}.

Accepted Answer

The MLE, no calculus allowed. Each observation has density 1/	heta on [0, 	heta], so L(	heta) = \frac{1}{	heta^n}\,\mathbf{1}\{	heta \geq \maxi Xi\}. For 	heta \geq \maxi Xi the likelihood 	heta^{-n} is strictly decreasing in 	heta; below the max it is zero. So the maximum sits at the boundary: \hat{	heta}{	ext{MLE}} = X{(n)} = \maxi Xi. Setting a derivative to zero would find nothing, the optimum is a corner, because the support depends on the parameter. The bias. The maximum's CDF is \mathbb{P}(X{(n)} \leq x) = (x/	heta)^n on [0,	heta], hence density n x^{n-1}/	heta^n and \math…

The biased MLE of Uniform(0, θ)

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