A sufficient statistic for a uniform upper edge

Question

You have independent observations X1, \dots, Xn from a Uniform(0, 	heta) distribution with unknown upper edge 	heta. Using the factorization idea, find a sufficient statistic for 	heta.

Accepted Answer

A twist: the parameter sits in the support. Each density is \frac{1}{	heta} for 0 \le xi \le 	heta and 0 otherwise. The dependence on 	heta hides not in an exponent but in a yes/no condition: every observation must be at most 	heta. Write that condition with an indicator \mathbf{1}\{\cdot\}. Factor with an indicator. The joint density is f(x; 	heta) = \prod{i=1}^n \frac{1}{	heta}\,\mathbf{1}\{0 \le xi \le 	heta\} = \frac{1}{	heta^n}\,\mathbf{1}\{\maxi xi \le 	heta\} \cdot \mathbf{1}\{\mini xi \ge 0\}. The key step: "every xi \le 	heta" is the same as "the largest xi \le 	heta." So …

A sufficient statistic for a uniform upper edge

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