Let X1,...,Xn be independently distributed according to the uniform distribution U(, + 1). (i) Uniformly most

Let X1,...,Xn be independently distributed according to the uniform distribution U(θ, θ + 1).

(i) Uniformly most accurate lower confidence bounds θ for θ at confidence level 1 − α exist and are given by

θ = max(X(1) − k, X(n) − 1), where X(1) = min(X1,...,Xn), X(n) = max(X1,...,Xn), and (1 − k)

n = α

(ii) The set C : x(n) − x(1) ≥ 1 − k is a relevant subset with Pθ(θ ≤ θ | C)=1 for all θ.

(iii) Determine the uniformly most accurate conditional lower confidence bounds θ(v) given the ancillary statistic V = X(n) − X(1) = v, and compare them with θ. [The conditional distribution of Y = X(1) given V = v is U(θ, θ + 1 − v).]

[Pratt (1961a), Barnard (1976).]