The maximum is sufficient for a uniform's endpoint

Question

Let X1, \dots, Xn \overset{iid}{\sim} 	ext{Uniform}(0, 	heta) with 	heta unknown. Show that the maximum T = \maxi Xi is sufficient for 	heta, and explain the role of the indicator function.

Accepted Answer

Fisher-Neyman factorization says T(X) is sufficient for 	heta if the joint density factors as g\big(T(x), 	heta\big)\,h(x) with h free of 	heta. Write the joint density. Each observation has density \frac{1}{	heta} on [0, 	heta] and 0 elsewhere. The product is nonzero only when every xi lies in [0, 	heta], which means \maxi xi \le 	heta and \mini xi \ge 0: f(x; 	heta) = \prod{i=1}^n \frac{1}{	heta}\,\mathbf{1}\{0 \le xi \le 	heta\} = 	heta^{-n}\,\mathbf{1}\{\maxi xi \le 	heta\}\,\mathbf{1}\{\mini xi \ge 0\}. Factor it. f(x; 	heta) = \underbrace{	heta^{-n}\,\mathbf{1}\{\maxi xi \…

The maximum is sufficient for a uniform's endpoint

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