Question

Prove Theorem 9.1.3

Short answer

This follows by determining the possible set of values of g₀ for which the null is not rejected on observing X = x.

Step-by-step solution

Statement of the theorem

Let X =(X_1,….,X_n) be a random sample from a distribution that depends on a parameter θ. Let g(θ) be a real-valued function, and suppose that for each possible value g₀ of g(θ), there is a level α₀ testδ_goof the hypotheses.

H_0go:g(θ) ≤ g₀ H_1go:g(θ) ≤ g₀

For each possible value x of X, define ω(x) by

ω(x) = { g₀ :δ_godoes not rejectH_0goif X=x is observed}

then

p[ g(θ₀)∈ ω(X)| θ = θ₀] ≥1-α₀ ∀θ₀∈Ω

Statement of the theorem

Let θ₀be an arbitrary element of Ω, and define g₀= g(θ₀). Becauseδ_gois a levelα₀ test, we know that.

P(δ_godoes not reject H_0go |θ = θ₀) ≥1-α₀

For each x, we know that g(θ₀)∈ ω(x) if and only if the testδ_godoes not reject H_0gowhen X=x is observed. It follows that

P[g(θ₀)∈ ω(X)| θ = θ₀] = PP(δ_godoes not reject H_0go | θ = θ₀)

For all θ₀∈Ω and thus the proof is complete.