Two edge statistics for a uniform band

Question

You have independent observations X1, \dots, Xn from a Uniform(a, b) distribution with both endpoints a and b unknown. Find a pair of statistics that are jointly sufficient for (a, b).

Accepted Answer

Both parameters live in the support. Each density is \frac{1}{b - a} when a \le xi \le b and 0 otherwise. The dependence on the parameters is carried by two yes/no conditions: every point is at least a, and every point is at most b. Write them as indicators. Factor with indicators. The joint density is f(x; a, b) = \frac{1}{(b - a)^n}\,\mathbf{1}\{	ext{all } xi \ge a\}\,\mathbf{1}\{	ext{all } xi \le b\}. Now the key simplifications: "all xi \ge a" is the same as "the smallest xi \ge a," and "all xi \le b" is the same as "the largest xi \le b." So f(x; a, b) = \frac{1}{(b - a)^n}\,\mathbf{1}…

Two edge statistics for a uniform band

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