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with tuning factors ι$_{1}$ $_{= (}$√$_{0}$$_{.}$$_{3}$$_{,}$ √$_{0}$$_{.}$$_{7}$$_{,}$ √$_{1}$$_{.}$$_{45) and}$ ι$_{2}$ $_{= (}$√$_{3}$$_{.}$$_{25}$$_{,}$ √$_{3}$$_{.}$$_{1}$$_{,}$ √$_{2}$$_{.}$$_{75) to achieve an}$ acceptance rate of approximately 33 percent. The inefficiency factors are low and trace plots, as displayed in Figure 3 for the 50th quantile, show quick convergence. Trace plots at the other two quantiles are similar. Computational time remains unchanged at approximately 160 seconds per 1 , 000 iterations. 

The results in Table 3 show that the posterior estimates for β are close to the true values (3 , − 7 , 5), posterior estimates of σ adjusts to capture the spread and posterior estimates of γ captures the skewness extremely well. Specifically, the posterior mean of γ is statistically not different from zero at the 25th quantile, so skewness is 1.64 and equals that of BQROR model. However, the skewness at the 50th (75th) quantile is 1.31 (0 . 16) compared to a skewness of 0 ( − 1 . 64) in the BQROR model. Hence, the FBQROR model correctly captures the positive skewness of the simulated data. Model comparison also points to the superiority of the FBQROR model as seen from Table 4. Conditional log-likelihood for the FBQROR model are higher than that of BQROR models at the 50th and 75th quantiles, but identical at the 25th quantile. AIC and BIC values suggest that there is extremely strong evidence to select the FBQROR model at the 50th and 75th quantiles, and some evidence to favor the BQROR model at the 25th quantile. Once again, the flexibility offered by the FBQROR model in terms of modeling the skewness helps to provide a better model fit compared to the rigid BQROR model. 

