8.6 Suppose that for a sample x1,...,xn coming from a distribution with density f(x|) we can construct the maximum likelihood function L(), with

8.6 Suppose that for a sample x1,...,xn coming from a distribution with density f(x|θ) we can construct the maximum likelihood function L(θ), with θ ∈ R. Suppose further that we can calculate ˆθ the MLE, that is, L(ˆθ) ≥ L(θ), for all θ ∈ R. Let h : R → Λ, where Λ denotes the codomain of the function, and let λ = h(θ) and

λˆ = h(ˆθ). For each λ, let Uλ be its pre-image through h, Uλ = {θ ∈ R : h(θ) = λ}, and for every λ define the function