Algorithm 4.

Step 1: Specify the values of $a_1$, $a_2$, $b_1$, $b_2$, and $r$, then compute $a(r)$ using Eq. (16) and compute $b^{+}(r)$ using Eq. (18)

Step 2: At the $i$ step

(a) Generate $u$ from uniform distribution with parameters 0 and $a(r)$, denoted as $U(0,a(r))$

(b) Generate $v$ from uniform distribution with parameters 0 and $b^{+}(r)$, denoted as $U(0,b^{+}(r))$

(c) Compute $\rho =\frac{v}{u^r}$

(d) If the value of $\rho$ is accepted, the set ${\beta }_{(i)}=\rho$ if $u\le [p(\beta |x){]}^{1/(r+1)}$; otherwise, repeat step (a)–step (c)

(e) Generate $\lambda$ from inverse gamma distribution with parameters $\frac{n}{2}+a_2$ and $\frac{1}{2}{\sum }_{j=1}^n\left(\frac{x_j}{{\beta }_{(i)}}+\frac{{\beta }_{(i)}}{x_j}-2\right)+b_2$, denoted as $IG\left(\frac{n}{2}+a_2,\frac{1}{2}{\sum }_{j=1}^n\left(\frac{x_j}{{\beta }_{(i)}}+\frac{{\beta }_{(i)}}{x_j}-2\right)+b_2\right)$ and compute ${\alpha }_{(i)}=\sqrt{\lambda }$

Step 3: Compute the posterior distribution of $\theta$, denoted as ${\theta }_{Baye}$, using Eq. (19)

Step 4: Repeat step 2 and step 3, a total M times and obtain an array of ${\theta }_{Baye}$’s

Step 5: Compute $L_{\theta .HPD}$ and $U_{\theta .HPD}$