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Statistics

Does the sample maximum pin down a uniform ceiling?

consistencyorder-statisticsuniform-distributionbias

Observations X1,…,XnX_1, \dots, X_n are i.i.d. Uniform(0,θ)(0, \theta) with unknown ceiling θ\theta. You estimate θ\theta by the largest value seen, θ^=X(n)=max⁡iXi\hat\theta = X_{(n)} = \max_i X_i, which is always strictly below θ\theta.

Is θ^\hat\theta a consistent estimator of θ\theta, despite always underestimating it?

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