Proof of Theorem 1. To prove Theorem 1, we follow the methodology to obtain reference priors proposed by Berger and Bernardo (1992a). In particular, we assume that the reader is familiar with both the notation and the methodology of Berger and Bernardo (1992a). This proof is divided in two parts. In the first part, we obtain the reference prior for the orderings (β,σ,p), (σ,β,p), and (σ,p,β). Because the proofs are analogous for each of these three orderings, in the first part we obtain the reference prior for the ordering (σ,β,p). In the second part, we obtain the reference prior for the orderings (β,p,σ), (p,β,σ), and (p,σ,β). Because the proofs are analogous for each of these three orderings, in the second part we obtain the reference prior for the ordering (p,β,σ).
Part 1. Consider the ordering θ=(σ,β,p).
After rearranging the Fisher information matrix H(θ) given in Equation (8) to conform to this ordering, the inverse of the Fisher information matrix becomes
Thus,
and S3=S(θ). Moreover, let . Thus,
and H3=H(θ).
Let h
j
be the n
j
×n
j
lower right corner of H
j
. Thus,
Let θ(1)=σ, θ(2)=β, and θ(3)=p. In addition, let θ[1]=θ(1)=σ, θ[2]=(θ(1),θ(2))=(σ,β), and θ[3]=(θ(1),θ(2),θ(3))=(σ,β,p). Moreover, let θ[∼1]=(θ(2),θ(3))=(β,p) and θ[∼2]=(θ(3))=p. Further, consider the following compact sets: for σ, ; for β, ; for p, .
Then,
where .
Now,
where
with
Hence,
Finally,
with
where
does not depend on θ=(σ,β,p).
Hence,
Thus,
Now take any point θ∗=(σ∗,β∗,p∗)∈ [ l−1,l]×[−l,l]k×[ 1,l]. Then, the reference prior for the ordering (σ,β,p) is
which is of the form (4).
Part 2. Consider the ordering θ=(p,β,σ).
After rearranging the Fisher information matrix H(θ) given in Equation (8) to conform to this ordering, the inverse of the Fisher information matrix becomes
Thus,
and S3=S(θ). Moreover, let . Thus,
and H3=H(θ).
Let h
j
be the n
j
×n
j
lower right corner of H
j
. Thus,
Let θ(1)=p, θ(2)=β, and θ(3)=σ. In addition, let θ[1]=θ(1)=p, θ[2]=(θ(1),θ(2))=(p,β), and θ[3]=(θ(1),θ(2),θ(3))=(p,β,σ). Moreover, let θ[∼1]=(θ(2),θ(3))=(β,σ) and θ[∼2]=(θ(3))=σ. Further, consider the following compact sets: for p, ; for β, ; for σ, .
Then,
Moreover,
where
Hence,
Further,
with
Hence,
where
Thus,
Now take any point θ∗=(p∗,β∗,σ∗)∈ [ 1,l]× [ −l,l]k× [ l−1,l]. Then, the reference prior for the ordering (p,β,σ) is
which is of the form (4).