**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

S(\theta )\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{H}^{-1}(\theta )\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\begin{array}{lll}\frac{{\sigma}^{2}}{\mathit{\text{np}}}\frac{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}& 0& \frac{\mathrm{\sigma p}}{n}\frac{1}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}\\ 0& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\sigma}^{2}{\left\{\phantom{\rule{0.3em}{0ex}}\Gamma \left({p}^{-1}\right)\Gamma \left(2-{p}^{-1}\right)\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}{x}_{i}{x}_{i}^{\prime}\right\}}^{\phantom{\rule{0.3em}{0ex}}-1}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}& 0\\ \frac{\mathrm{\sigma p}}{n}\frac{1}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}& 0& \frac{{p}^{3}}{n}\frac{1}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}\end{array}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right]\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}.

Thus,

{S}_{1}=\frac{{\sigma}^{2}}{\mathit{\text{np}}}\frac{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)}{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)-1},

{S}_{2}=\left[\begin{array}{ll}\frac{{\sigma}^{2}}{\mathit{\text{np}}}\frac{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}& 0\\ 0& {\sigma}^{2}{\left\{\Gamma \left({p}^{-1}\right)\Gamma \left(2-{p}^{-1}\right)\sum _{i=1}^{n}{x}_{i}{x}_{i}^{\prime}\right\}}^{-1}\end{array}\right],

and *S*_{3}=*S*(*θ*). Moreover, let {H}_{j}={S}_{j}^{-1}. Thus,

{H}_{1}=\frac{\mathit{\text{np}}}{{\sigma}^{2}}\frac{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)-1}{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)},

{H}_{2}=\left[\begin{array}{ll}\frac{\mathit{\text{np}}}{{\sigma}^{2}}\frac{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})}& 0\\ 0& {\sigma}^{-2}\Gamma \left({p}^{-1}\right)\Gamma \left(2-{p}^{-1}\right)\sum _{i=1}^{n}{x}_{i}{x}_{i}^{\prime}\end{array}\right],

and *H*_{3}=*H*(*θ*).

Let *h*_{
j
} be the *n*_{
j
}×*n*_{
j
} lower right corner of *H*_{
j
}. Thus,

\begin{array}{ll}{h}_{1}& =\frac{\mathit{\text{np}}}{{\sigma}^{2}}\frac{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)-1}{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)},\phantom{\rule{2em}{0ex}}\\ {h}_{2}& ={\sigma}^{-2}\Gamma \left({p}^{-1}\right)\Gamma \left(2-{p}^{-1}\right)\sum _{i=1}^{n}{x}_{i}{x}_{i}^{\prime},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\\ {h}_{3}& =n{p}^{-3}\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right).\phantom{\rule{2em}{0ex}}\end{array}

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*, {\Theta}_{(1)}^{l}=\phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}{l}^{-1},l]; for *β*, {\Theta}_{(2)}^{l}=\phantom{\rule{2.77626pt}{0ex}}{[\phantom{\rule{0.3em}{0ex}}-l,l]}^{k}; for *p*, {\Theta}_{(3)}^{l}=\phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}1,l].

Then,

\begin{array}{l}{\pi}_{3}^{l}(p\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}\sigma ,\beta )={\pi}_{3}^{l}\left({\theta}_{[\sim 2]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[2\right]}\right)\\ \phantom{\rule{5.8em}{0ex}}=\frac{|{h}_{3}(\theta ){|}^{1/2}{\mathbf{1}}_{{\Theta}_{(3)}^{l}}\left({\theta}_{(3)}\right)}{{\int}_{{\Theta}_{(3)}^{l}}|{h}_{3}(\theta ){|}^{1/2}d{\theta}_{(3)}}\\ \phantom{\rule{5.8em}{0ex}}=\frac{{\left\{n{p}^{-3}\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)\right\}}^{1/2}{\mathbf{1}}_{[1,l]}(p)}{{\int}_{1}^{l}{\left\{n{p}^{-3}\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)\right\}}^{1/2}\mathit{\text{dp}}}\\ \phantom{\rule{5.8em}{0ex}}={\left\{{c}_{1}(l)\right\}}^{-1}{p}^{-3/2}{\left(1+{p}^{-1}\right)}^{1/2}{\left\{{\Psi}^{\prime}\left(1+{p}^{-1}\right)\right\}}^{1/2}{\mathbf{1}}_{[1,l]}(p),\end{array}

where {c}_{1}(l)={\int}_{1}^{l}{p}^{-3/2}{(1+{p}^{-1})}^{1/2}{\left\{{\Psi}^{\prime}(1+{p}^{-1})\right\}}^{1/2}\mathit{\text{dp}}.

Now,

\begin{array}{l}{\pi}_{2}^{l}(\beta ,p\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}\sigma )={\pi}_{2}^{l}\left({\theta}_{[\sim 1]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[1\right]}\right)\\ \phantom{\rule{6em}{0ex}}=\frac{{\pi}_{3}^{l}\left({\theta}_{[\sim 2]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[2\right]}\right)exp\left\{0.5\underset{2}{\overset{l}{E}}\left[log|{h}_{2}(\theta )|\left|\right.{\theta}_{\left[2\right]}\right]\right\}{\mathbf{1}}_{{\Theta}_{(2)}^{l}}\left({\theta}_{(2)}\right)}{{\int}_{{\Theta}_{(2)}^{l}}exp\left\{0.5{E}_{2}^{l}\left[log|{h}_{2}(\theta )|\left|\right.{\theta}_{\left[2\right]}\right]\right\}d{\theta}_{(2)}},\end{array}

where

\begin{array}{l}{E}_{2}^{l}\left[log|{h}_{2}(\theta )|\left|\right.{\theta}_{\left[2\right]}\right]={\int}_{{\Theta}_{(3)}^{l}}log|{h}_{2}(\theta )|{\pi}_{3}^{l}({\theta}_{[\sim 2]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[2\right]})d{\theta}_{[\sim 2]}\\ \phantom{\rule{9.7em}{0ex}}={\int}_{1}^{l}\left\{\phantom{\rule{0.3em}{0ex}}-2k\phantom{\rule{0.3em}{0ex}}log\sigma \phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}klog\Gamma \phantom{\rule{0.3em}{0ex}}\left({p}^{-1}\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}klog\Gamma \phantom{\rule{0.3em}{0ex}}\left(2-{p}^{-1}\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}log\left|\right.\sum _{i=1}^{n}{x}_{i}\underset{i}{\overset{\prime}{x}}\left|\right.\phantom{\rule{0.3em}{0ex}}\right\}\\ \phantom{\rule{12.3em}{0ex}}{\left\{{c}_{1}(l)\right\}}^{-1}{p}^{-3/2}{\left(1+{p}^{-1}\right)}^{1/2}{\left\{{\Psi}^{\prime}\left(1+{p}^{-1}\right)\right\}}^{1/2}\mathit{\text{dp}}\\ \phantom{\rule{9.7em}{0ex}}=-2klog\sigma +{c}_{2}(l),\end{array}

with

\begin{array}{l}{c}_{2}(l)={\left\{{c}_{1}(l)\right\}}^{-1}{\int}_{1}^{l}\left\{klog\Gamma \left({p}^{-1}\right)+klog\Gamma \left(2-{p}^{-1}\right)+log\left|\right.\sum _{i=1}^{n}{x}_{i}\underset{i}{\overset{\prime}{x}}\left|\right.\right\}\\ \phantom{\rule{9.7em}{0ex}}{p}^{-3/2}{\left(1+{p}^{-1}\right)}^{1/2}{\left\{{\Psi}^{\prime}\left(1+{p}^{-1}\right)\right\}}^{1/2}\mathrm{dp.}\end{array}

Hence,

\begin{array}{l}{\pi}_{2}^{l}(\beta ,p\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}\sigma )=\frac{{\pi}_{3}^{l}(p|\sigma ,\beta )exp\left\{0.5[-2klog\sigma +{c}_{2}(l)]\right\}{\mathbf{1}}_{{[-l,l]}^{k}}(\beta )}{{\int}_{{[-l,l]}^{k}}exp\left\{0.5[-2klog\sigma +{c}_{2}(l)]\right\}\mathrm{d\beta}}\\ \phantom{\rule{6em}{0ex}}={\pi}_{3}^{l}(p|\sigma ,\beta ){(2l)}^{-k}{\mathbf{1}}_{{[-l,l]}^{k}}(\beta ).\end{array}

Finally,

\begin{array}{l}{\pi}_{1}^{l}(\sigma ,\beta ,p)={\pi}_{1}^{l}\left({\theta}_{[\sim 0]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[0\right]}\right)\\ \phantom{\rule{5em}{0ex}}=\frac{{\pi}_{2}^{l}\left({\theta}_{[\sim 1]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[1\right]}\right)exp\left\{0.5\underset{1}{\overset{l}{E}}\left[log|{h}_{1}(\theta )|\left|\right.{\theta}_{\left[1\right]}\right]\right\}{\mathbf{1}}_{{\Theta}_{(1)}^{l}}\left({\theta}_{(1)}\right)}{{\int}_{{\Theta}_{(1)}^{l}}exp\left\{0.5\underset{1}{\overset{l}{E}}\left[log|{h}_{1}(\theta )|\left|\right.{\theta}_{\left[1\right]}\right]\right\}d{\theta}_{(1)}},\end{array}

with

\begin{array}{l}{E}_{1}^{l}\left[log|{h}_{1}(\theta )|\left|\right.{\theta}_{\left[1\right]}\right]={\int}_{{[-l,l]}^{k}}{\int}_{1}^{l}log\left\{\frac{\mathit{\text{np}}}{{\sigma}^{2}}\frac{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)-1}{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)}\right\}\underset{2}{\overset{l}{\pi}}(\beta ,p\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}\sigma )\mathrm{dpd\beta}\\ \phantom{\rule{10em}{0ex}}=-2log\sigma +{c}_{3}(l),\end{array}

where

{c}_{3}(l)={\int}_{{[-l,l]}^{k}}{\int}_{1}^{l}log\left\{\mathit{\text{np}}\frac{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)-1}{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)}\right\}\underset{2}{\overset{l}{\pi}}(\beta ,p\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}\sigma )\mathrm{dpd\beta}

does not depend on *θ*=(*σ*,*β*,*p*).

Hence,

\begin{array}{l}{\pi}_{1}^{l}(\sigma ,\beta ,p)=\frac{{\pi}_{2}^{l}(\beta ,p|\sigma )exp\{0.5[\phantom{\rule{0.3em}{0ex}}2log\sigma +{c}_{3}(l)]\}{\mathbf{1}}_{({l}^{-1},l)}(\sigma )}{{\int}_{{l}^{-1}}^{l}exp\left\{0.5\right[\phantom{\rule{0.3em}{0ex}}2log\sigma +{c}_{3}(l)\left]\right\}\mathrm{d\sigma}}\\ \phantom{\rule{5em}{0ex}}=\frac{{\pi}_{2}^{l}(\beta ,p|\sigma ){\sigma}^{-1}{\mathbf{1}}_{({l}^{-1},l)}(\sigma )}{2logl}\end{array}

Thus,

\begin{array}{ll}{\pi}_{1}^{l}(\sigma ,\beta ,p)=& \phantom{\rule{2.56804pt}{0ex}}{\sigma}^{-1}{p}^{-3/2}{\left(1+{p}^{-1}\right)}^{1/2}{\left\{{\Psi}^{\prime}\left(1+{p}^{-1}\right)\right\}}^{1/2}\phantom{\rule{2em}{0ex}}\\ \times \phantom{\rule{2.56804pt}{0ex}}{\{{c}_{1}(l)\}}^{-1}{(2l)}^{-k}{(2logl)}^{-1}{\mathbf{1}}_{[{l}^{-1},l]}(\sigma ){\mathbf{1}}_{{[-l,l]}^{k}}(\beta ){\mathbf{1}}_{[1,l]}(p).\phantom{\rule{2em}{0ex}}\end{array}

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

\begin{array}{l}\pi (\sigma ,\beta ,p)\propto {\text{lim}}_{l\to \infty}\frac{{\pi}_{1}^{l}(\sigma ,\beta ,p)}{{\pi}_{1}^{l}({\sigma}^{\ast},{\beta}^{\ast},{p}^{\ast})}\\ \phantom{\rule{4.3em}{0ex}}={\sigma}^{-1}{p}^{-3/2}{\left(1+{p}^{-1}\right)}^{1/2}{\left\{{\Psi}^{\prime}\left(1+{p}^{-1}\right)\right\}}^{1/2},\end{array}

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

\phantom{\rule{-20.0pt}{0ex}}S(\theta )\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{H}^{-1}(\theta )\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\begin{array}{lll}\frac{{p}^{3}}{n}\frac{1}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}& \phantom{\rule{10em}{0ex}}0& \frac{\mathrm{\sigma p}}{n}\frac{1}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}\\ \phantom{\rule{7em}{0ex}}0& \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\sigma}^{2}{\left\{\Gamma \phantom{\rule{0.3em}{0ex}}\left({p}^{-1}\right)\Gamma \left(2-{p}^{-1}\right)\sum _{i=1}^{n}{x}_{i}{x}_{i}^{\prime}\right\}}^{-1}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}& \phantom{\rule{7em}{0ex}}0\\ \frac{\mathrm{\sigma p}}{n}\frac{1}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}& \phantom{\rule{10em}{0ex}}0& \frac{{\sigma}^{2}}{\mathit{\text{np}}}\frac{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})}{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1}\end{array}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\right]\phantom{\rule{0.3em}{0ex}},

Thus,

{S}_{1}=\frac{{p}^{3}}{n}\frac{1}{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)-1},

{S}_{2}=\left[\begin{array}{ll}\frac{{p}^{3}}{n}\frac{1}{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)-1}& \phantom{\rule{9em}{0ex}}0\\ \phantom{\rule{7em}{0ex}}0& {\sigma}^{2}{\left\{\Gamma ({p}^{-1})\Gamma \left(2-{p}^{-1}\right)\sum _{i=1}^{n}{x}_{i}{x}_{i}^{\prime}\right\}}^{-1}\end{array}\right],

and *S*_{3}=*S*(*θ*). Moreover, let {H}_{j}={S}_{j}^{-1}. Thus,

{H}_{1}=n{p}^{-3}\{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\},

{H}_{2}=\left[\begin{array}{ll}n{p}^{-3}\{\left(1+{p}^{-1}\right){\Psi}^{\prime}\left(1+{p}^{-1}\right)-1\}& \phantom{\rule{8em}{0ex}}0\\ \phantom{\rule{8.6em}{0ex}}0& {\sigma}^{-2}\Gamma \left({p}^{-1}\right)\Gamma \left(2-{p}^{-1}\right)\sum _{i=1}^{n}{x}_{i}{x}_{i}^{\prime}\end{array}\right],

and *H*_{3}=*H*(*θ*).

Let *h*_{
j
} be the *n*_{
j
}×*n*_{
j
} lower right corner of *H*_{
j
}. Thus,

\phantom{\rule{-19.0pt}{0ex}}{h}_{1}=n{p}^{-3}\{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\},\phantom{\rule{1em}{0ex}}{h}_{2}={\sigma}^{-2}\Gamma \left({p}^{-1}\right)\Gamma \left(2-{p}^{-1}\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{n}{x}_{i}{x}_{i}^{\prime},\text{and}{h}_{3}=\mathit{\text{np}}{\sigma}^{-2}.

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*, {\Theta}_{(1)}^{l}=\phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}1,l]; for *β*, {\Theta}_{(2)}^{l}=\phantom{\rule{2.77626pt}{0ex}}{[\phantom{\rule{0.3em}{0ex}}-l,l]}^{k}; for *σ*, {\Theta}_{(3)}^{l}=\phantom{\rule{2.77626pt}{0ex}}[\phantom{\rule{0.3em}{0ex}}{l}^{-1},l].

Then,

\begin{array}{l}{\pi}_{3}^{l}(\sigma \phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}p,\beta )={\pi}_{3}^{l}\left({\theta}_{[\sim 2]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[2\right]}\right)\\ \phantom{\rule{5.7em}{0ex}}=\frac{|{h}_{3}(\theta ){|}^{1/2}{\mathbf{1}}_{{\Theta}_{(3)}^{l}}\left({\theta}_{(3)}\right)}{{\int}_{{\Theta}_{(3)}^{l}}|{h}_{3}(\theta ){|}^{1/2}d{\theta}_{(3)}}\\ \phantom{\rule{5.7em}{0ex}}=\frac{{\left\{\mathit{\text{np}}{\sigma}^{-2}\right\}}^{1/2}{\mathbf{1}}_{[{l}^{-1},l]}(\sigma )}{{\int}_{{l}^{-1}}^{l}{\left\{\mathit{\text{np}}{\sigma}^{-2}\right\}}^{1/2}\mathrm{d\sigma}}\\ \phantom{\rule{5.7em}{0ex}}={\sigma}^{-1}{(2logl)}^{-1}{\mathbf{1}}_{[{l}^{-1},l]}(\sigma ).\end{array}

Moreover,

\begin{array}{l}{\pi}_{2}^{l}(\beta ,p\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}\sigma )={\pi}_{2}^{l}({\theta}_{[\sim 1]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[1\right]})\\ \phantom{\rule{5.7em}{0ex}}=\frac{{\pi}_{3}^{l}({\theta}_{[\sim 2]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[2\right]})exp\left\{0.5\underset{2}{\overset{l}{E}}\left[log|{h}_{2}(\theta )|\left|\right.{\theta}_{\left[2\right]}\right]\right\}{\mathbf{1}}_{{\Theta}_{(2)}^{l}}({\theta}_{(2)})}{{\int}_{{\Theta}_{(2)}^{l}}exp\left\{0.5\underset{2}{\overset{l}{E}}\left[log|{h}_{2}(\theta )|\left|\right.{\theta}_{\left[2\right]}\right]\right\}d{\theta}_{(2)}},\end{array}

where

\begin{array}{l}{E}_{2}^{l}\left[log|{h}_{2}(\theta )|\left|\right.{\theta}_{\left[2\right]}\right]={\int}_{{\Theta}_{(3)}^{l}}log|{h}_{2}(\theta )|{\pi}_{3}^{l}({\theta}_{[\sim 2]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[2\right]})d{\theta}_{[\sim 2]}\\ \phantom{\rule{9.7em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{\int}_{{l}^{-1}}^{l}\phantom{\rule{0.3em}{0ex}}\left\{\phantom{\rule{0.3em}{0ex}}-2klog\sigma +klog\Gamma ({p}^{-1})+klog\Gamma (2-{p}^{-1})+log\left|\right.\sum _{i=1}^{n}{x}_{i}\underset{i}{\overset{\prime}{x}}\left|\right.\phantom{\rule{0.3em}{0ex}}\right\}\\ \phantom{\rule{15em}{0ex}}{\sigma}^{-1}{(2logl)}^{-1}\mathrm{d\sigma}\\ \phantom{\rule{9.7em}{0ex}}={c}_{1}(l,p),\text{which does not depend on}\phantom{\rule{1em}{0ex}}\beta \text{.}\end{array}

Hence,

\begin{array}{l}{\pi}_{2}^{l}(\beta ,\sigma \phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}p)=\frac{{\pi}_{3}^{l}(\sigma |p,\beta )exp\left\{0.5{c}_{1}(l,p)\right\}{\mathbf{1}}_{{[-l,l]}^{k}}(\beta )}{{\int}_{{[-l,l]}^{k}}exp\left\{0.5{c}_{1}(l,p)\right\}\mathrm{d\beta}}\\ \phantom{\rule{5.7em}{0ex}}={\pi}_{3}^{l}(\sigma |p,\beta ){(2l)}^{-k}{\mathbf{1}}_{{[-l,l]}^{k}}(\beta ).\end{array}

Further,

\begin{array}{l}{\pi}_{1}^{l}(p,\beta ,\sigma )={\pi}_{1}^{l}({\theta}_{[\sim 0]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[0\right]})\\ \phantom{\rule{5em}{0ex}}=\frac{{\pi}_{2}^{l}({\theta}_{[\sim 1]}\phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}{\theta}_{\left[1\right]})exp\left\{0.5\underset{1}{\overset{l}{E}}\left[log|{h}_{1}(\theta )|\left|\right.{\theta}_{\left[1\right]}\right]\right\}{\mathbf{1}}_{{\Theta}_{(1)}^{l}}({\theta}_{(1)})}{{\int}_{{\Theta}_{(1)}^{l}}exp\left\{0.5\underset{1}{\overset{l}{E}}\left[log|{h}_{1}(\theta )|\left|\right.{\theta}_{\left[1\right]}\right]\right\}d{\theta}_{(1)}},\end{array}

with

\begin{array}{l}{E}_{1}^{l}\left[log|{h}_{1}(\theta )|\left|\right.{\theta}_{\left[1\right]}\right]=\phantom{\rule{0.3em}{0ex}}{\int}_{{[-l,l]}^{k}}{\int}_{{l}^{-1}}^{l}\phantom{\rule{0.3em}{0ex}}log\left[n{p}^{-3}\{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\}\right]\underset{2}{\overset{l}{\pi}}(\beta ,\sigma \phantom{\rule{2.77626pt}{0ex}}|\phantom{\rule{2.77626pt}{0ex}}p)\mathrm{d\sigma d\beta}\\ \phantom{\rule{9.7em}{0ex}}=log\left[n{p}^{-3}\{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\}\right].\end{array}

Hence,

\begin{array}{lcr}{\pi}_{1}^{l}(\sigma ,\beta ,p)& =& \frac{{\pi}_{2}^{l}(\beta ,\sigma |p)exp\{0.5log\left[n{p}^{-3}\{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\}\right]\}{\mathbf{1}}_{(1,l)}(p)}{{\int}_{1}^{l}exp\{0.5log\left[n{p}^{-3}\{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\}\right]\}\mathit{\text{dp}}}\\ =& {\pi}_{2}^{l}(\beta ,p|\sigma ){p}^{-3/2}{\{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\}}^{1/2}{c}_{2}(l){\mathbf{1}}_{[1,l]}(p),\end{array}

where

{\{{c}_{2}(l)\}}^{-1}={\int}_{1}^{l}exp\{0.5log\left[n{p}^{-3}\{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\}\right]\}\mathrm{dp.}

Thus,

\phantom{\rule{-21.0pt}{0ex}}{\pi}_{1}^{l}(p,\beta ,\sigma )\phantom{\rule{0.3em}{0ex}}={\sigma}^{-1}{p}^{-3/2}{\{\phantom{\rule{0.3em}{0ex}}(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\phantom{\rule{0.3em}{0ex}}\}}^{1/2}{c}_{2}(l){(2l)}^{-k}{(2logl)}^{-1}\phantom{\rule{0.3em}{0ex}}{\mathbf{1}}_{[1,l]}(p)\phantom{\rule{0.3em}{0ex}}{\mathbf{1}}_{{[-l,l]}^{k}}(\beta )\phantom{\rule{0.3em}{0ex}}{\mathbf{1}}_{[{l}^{-1},l]}(\sigma ).

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

\begin{array}{l}\pi (p,\beta ,\sigma )\propto {\text{lim}}_{l\to \infty}\frac{{\pi}_{1}^{l}(p,\beta ,\sigma )}{{\pi}_{1}^{l}({p}^{\ast},{\beta}^{\ast},{\sigma}^{\ast})}\\ \phantom{\rule{4em}{0ex}}={\sigma}^{-1}{p}^{-3/2}{\{(1+{p}^{-1}){\Psi}^{\prime}(1+{p}^{-1})-1\}}^{1/2},\end{array}

which is of the form (4).