Reference no: EM131851885

Let X₁, ..., X_n be i.i.d. binary random variables with P(X_i = 1) = θ, where θ ∈ (0, 1) is unknown and n is an even integer. Consider the problem of testing H₀; θ ≤ 0.5 versus H₁; θ > 0.5 with action space {0, 1} (0 means H₀ is accepted and 1 means H₁ is accepted). Let the loss function be L(θ, a) = 0 if H_j is true and a = j, j = 0, 1; L(θ, 0) = C₀ when θ > 0.5; and L(θ, 1) = C₁ when θ ≤ 0.5, where C₀ > C₁ > 0 are some constants. Calculate the risk function of the following randomized test (decision rule):