Question

Suppose that a single observation is to be drawn from the following p.d.f. : f(x|θ) = (θ + 1)x^−(θ+2)), 1 ≤ x < ∞, where the value of θ is unknown. Suppose that the following hypotheses are to be tested: null hypothesis H₀ : θ = 0 versus simple alternative H1 : θ = 1

(a) Determine (in terms of the rejection region) the Bayesian test procedure, corresponding to prior probability P(H₀) = 1/2 and posterior odds 1 : 1; 3 : 1 (substantial evidence) and 10 : 1 (strong evidence).

(b) Determine (in terms of the rejection region) the Bayesian test procedure, corresponding to prior probability P(H₀) = 1/3 and posterior odds 1 : 1.

(c) Determine (in terms of the rejection region) the Bayesian test procedure, corresponding to prior probability P(H₀) = 2/3 and posterior odds 1 : 1.

Answer 1