Let x1 xn be iid with the lebesgue pdf x-1i01x where 0

Question

Let X₁, ..., X_n be i.i.d. with the Lebesgue p.d.f. θx^θ-1I_(0,1)(x), where θ > 0 is unknown.

(a) Construct a confidence interval for θ with confidence coefficient 1 - α, using a sufficient statistic.

(b) Discuss whether the confidence interval obtained in (a) has the shortest length within a class of confidence intervals.

(c) Discuss whether the confidence interval obtained in (a) is UMAU.