(i) Let X1,..., Xn be a sample from the uniform distributionU(0, ), 0 < < ,...

(i) Let X1,..., Xn be a sample from the uniform distributionU(0, θ), 0 < θ < ∞, and let T = max(X1,..., Xn). Show that T is sufficient, once by using the definition of sufficiency and once by using the factorization criterion and assuming the existence of statistics Yi satisfying (1.17)–(1.19).

(ii) Let X1,..., Xn be a sample from the exponential distribution E

(a,

b) with density (1/b)e−(x−a)/b when x ≥ a (−∞ < a < ∞, 0 < b). Use the factorization criterion to prove that (min(X1,..., Xn),n i=1 Xi) is sufficient for

a, b, assuming the existence of statistics Yi satisfying (1.17)–(1.19).