Table 1: Posterior mean ( mean ), standard deviation ( std ) and inefficiency factor ( if ) of the FBQROR model parameters in Simulation Study 1. 25th quantile 50th quantile 75th quantile parameters mean std if mean std if mean std if β 1 1 . 07 0 . 32 2 . 94 2 . 17 0 . 31 2 . 66 3 . 23 0 . 35 4 . 91 β 2 − 3 . 22 0 . 50 4 . 02 − 3 . 14 0 . 48 3 . 67 − 3 . 13 0 . 47 4 . 72 β 3 3 . 93 0 . 53 4 . 08 3 . 86 0 . 51 3 . 82 3 . 90 0 . 50 5 . 28 σ 0 . 64 0 . 10 3 . 38 0 . 75 0 . 09 4 . 30 0 . 60 0 . 09 4 . 10 γ 1 . 14 0 . 27 2 . 55 − 0 . 06 0 . 17 3 . 76 − 1 . 18 0 . 24 2 . 93 δ 1 0 . 73 0 . 14 4 . 75 0 . 71 0 . 15 4 . 84 0 . 67 0 . 14 5 . 87 Inefficiency factors for all the model parameters are low which imply small correlation in MCMC draws and trace plots of the MCMC iterations, as exhibited in Figure 2 for the 25th quantile, display quick convergence. Trace plots for the other two quantiles are similar and have not been shown for the sake of brevity. The sampler is reasonably quick and takes approximately 160 seconds per 1 , 000 iterations. The results in Table 1 show that the posterior means for β are close to the true parameter values, posterior mean of σ adjusts the scale of the distribution and the posterior mean of δ$_{1}$ yields a value of ξ$_{3}$ close to 4, the true value used to generate the data. The posterior mean of γ at p$_{0}$ = (0 . 25 , 0 . 50 , 0 . 75) are (1 . 14 , − 0 . 06 , − 1 . 18), which corresponds to a skewness of (0 . 01 , 0 . 20 , 0 . 04), respectively. Note that the posterior mean of γ at p$_{0}$ = 0 . 50 is statistically equivalent to zero. These skewness values imply that the (latent) response variable is about symmetric at all the considered quantiles, which is reassuring since our data was generated from a symmetric distribution. In contrast, if we estimate the BQROR model which is more rigid, the corresponding skewness values are (1 . 64 , 0 , − 1 . 64). Hence, the BQROR model fails to accommodate the symmetric characteristic of the data at the 25th and 75th quantiles. We next investigate model fitness at different quantiles since various choices of quantile p$_{0}$ may be interpreted as corresponding to a different link function. Besides, to illustrate the practical utility of FBQROR models, we also estimate the BQROR model using a mod- ification of Algorithm 1 in Rahman (2016) – by fixing the second cut-point and introduc- ing a scale parameter in the model. Table 2 presents the conditional log-likelihood, the 14