A show that the set of risk points of all decision rules is

Question

Let X be a binary observation with P(X = 1) = θ₁ or θ₂, where 0 <>₁ <>₂ < 1="" are="" known="" values.="" consider="" the="" estimation="" of="" θ="" with="" action="" space="">₁, a₂} and loss function L(θ_i, a_j) = l_ij, where l₂₁ ≥ l₁₂ > l₁₁ = l₂₂= 0. For a decision rule δ(X), the vector (R_δ(θ₁), R_δ(θ₂)) is defined to be its risk point.

(a) Show that the set of risk points of all decision rules is the convex hull of the set of risk points of all nonrandomized rules.

(b) Find a minimax rule.

(c) Let Π be a distribution on {θ₁, θ₂}. Obtain the class of all Bayes rules w.r.t. Π. Discuss when there is a unique Bayes rule.