(Berger, 1985, pg. 134) Let X be such that p(x | θ) = exp[−(x − θ)], x ≥…

(Berger, 1985, pg. 134) Let X be such that p(x | θ) = exp[−(x − θ)], x ≥ θ and assume the prior p(θ) ∝ (1 + θ 2 ) −1 , θ ≥ 0.

(a) Prove that the posterior distribution of θ given x is monotonically increasing and that the mode of θ is x.

(b) Show that the 100(1 − α)% HPD interval for θ must have the form [c(α), x], where c(α) is such that P[c(α) ≤ θ ≤ x | x] = 1 − α.

(c) Obtain the posterior density of η = exp(θ) and prove it is a monotonically increasing function of η.

(d) Show that the 100(1 − α)% HPD interval for η must have the form [1, d(α)], where d(α) is such that P(1 ≤ η ≤ d(α) | x) = 1 − α.

(e) Show that the credibility interval in (d) implies a 100(1 − α)% lowest posterior density interval for θ.