Lt x1 xn n 1 be iid from the exponential distribution e

Question

Let X₁, ..., X_n (n > 1) be i.i.d. from the exponential distribution E(θ, θ), where θ > 0 is unknown.

(a) Show that both X/Θ and X₍₁₎/θ are pivotal quantities, where X is the sample mean and X₍₁₎ is the smallest order statistic.

(b) Obtain confidence intervals (with confidence coefficient 1 - α) for θ based on the two pivotal quantities in (a).

(c) Discuss which confidence interval in (b) is better in terms of the length