# A New Quantile Regression for Modeling Bounded Data under a Unit Birnbaum–Saunders Distribution with Applications in Medicine and Politics

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## Abstract

**:**

## 1. Introduction

## 2. The Unit Birnbaum–Saunders Distribution

## 3. The UBS Quantile Regression Model

- (i)
- [Logit] $g\left({\mu}_{i}\right)=log\left(\right)open="["\; close="]">{\mu}_{i}/(1-{\mu}_{i})$;
- (ii)
- [Probit] $g\left({\mu}_{i}\right)={\Phi}^{-1}\left({\mu}_{i}\right)$;
- (iii)
- [Complementary log-log] $g\left({\mu}_{i}\right)=log\left(\right)open="["\; close="]">-log(1-{\mu}_{i})$;
- (iv)
- [Log-log] $g\left({\mu}_{i}\right)=log[-log\left({\mu}_{i}\right)]$; and
- (v)
- [Cauchit] $g\left({\mu}_{i}\right)=tan\left[\pi (\mu -0.5)\right]$;

`R`software named

`unitBSQuantReg`by means of its function

`unitBSQuantReg(formula, tau, data, link,…)`. For model checking, the function

`hnp(object, nsim = 99, halfnormal = TRUE, plot = TRUE, level = 0.95,`

`resid.type = c(“cox-snell”, “quantile”))`is available, which provides half-normal probability plots with simulated envelopes. A version of the

`unitBSQuantReg`package is available at https://github.com/AndrMenezes/unitBSQuantReg (accessed on 9 April 2021).

## 4. A Monte Carlo Simulation Study

- (i)
- $\mathrm{logit}\left({\mu}_{i}\right)={\delta}_{0}+{\delta}_{1}{z}_{i1}$ for ${\delta}_{0}=1.0$, ${\delta}_{1}=2.0$, with ${z}_{i1}\sim \mathrm{N}(0,1)$; and
- (ii)
- $\mathrm{logit}\left({\mu}_{i}\right)={\delta}_{0}+{\delta}_{1}{z}_{i1}+{\delta}_{2}{z}_{i2}$ for ${\delta}_{0}=1.0$, ${\delta}_{1}=1.0$, ${\delta}_{2}=2.0$, ${z}_{i1}\sim \mathrm{N}(0,1)$, with ${z}_{i2}\sim \mathrm{N}(0,1)$.

`SAS Data-Step`, while parameter estimates were obtained by the quasi-Newton method in

`SAS PROC NLMIXED`. The values of the response variable, given n, $\tau $, $\alpha $ and the covariate(s), are generated from the quantile function stated as

- (i)
- Bias$\left(\right)open="("\; close=")">\widehat{\varrho}$;
- (ii)
- RMSE$\left({\widehat{\varrho}}_{i}\right)={\left[(1/B){\sum}_{i=1}^{B}{({\widehat{\varrho}}_{i}-\varrho )}^{2}\right]}^{1/2}$; and
- (iii)
- CP${}_{95\%}\left(\right)open="("\; close=")">{\widehat{\varrho}}_{i}$, where $\varrho =\alpha ,{\delta}_{0},{\delta}_{1}$, or ${\delta}_{2}$, I is the indicator function, and $\mathrm{SE}\left({\widehat{\varrho}}_{i}\right)$ is the corresponding estimated SE.

## 5. Empirical Application with Data from Political Sciences

`baquantreg`[53] from

`R`and was considered the human development index (HDI) in 2010 as covariate. We call this set “vote data” and assume the regression structure for ${\mu}_{i}$ formulated as $\mathrm{logit}\left({\mu}_{i}\right)={\delta}_{0}+{\delta}_{1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{HDI}}_{i}$, for $i=1,\dots ,74$.

`unitBSQuantReg`package using the instructions:

```
install.packages(‘‘remotes’’)
remotes::install_github(‘‘AndrMenezes/unitBSQuantReg’’)
library(unitBSQuantReg)
data(BrazilElection2014, package = ‘‘baquantreg’’)
ind <- which(BrazilElection2014$UF_name == ‘‘SE’’)
data.se <- BrazilElection2014[ind, c(‘‘percVotes", ‘‘HDI’’)]
taus <- c(0.10, 0.25, 0.50, 0.75, 0.90)
fits <- lapply(taus, function(TAU)
unitBSQuantReg(percVotes ~ HDI, data = data.se, tau = TAU))
lapply(fits, coef)
```

`unitquantreg`package, which is under development.

## 6. Empirical Application with Data from Sports Medicine

## 7. Conclusions, Limitations and Future Investigation

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Other Distributions for Quantile Regression

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**Figure 1.**PDFs of the unit UBS distribution for the indicated values of $\mu $, $\alpha $ and $\tau $.

**Figure 2.**Behaviors of the mean, variance, coefficient of skewness and coefficient of kurtosis for the indicated values of $\alpha $ and $\tau $ in function of $\mu $.

**Figure 3.**Quantile regression fit plot (

**left**) and estimated quantile process plots for ${\delta}_{0}$ (

**center left**), ${\delta}_{1}$ (

**center right**) and $\alpha $ (

**right**) with vote data.

**Figure 4.**Half-normal plots with envelopes of Cox–Snell (first row) and randomized quantile (second row) residuals for the indicated $\tau $ quantile level with vote data.

**Figure 5.**Estimated quantile process plot for the indicated ${\delta}_{j}$, with $j=0,1,\dots ,5$, and $\alpha $ (second row right) with arm data.

**Figure 6.**Half-normal plots with envelopes of Cox–Snell (first row) and randomized quantile (second row) residuals for the indicated $\tau $ quantile level with arm data.

**Table 1.**Empirical bias, RMSE and $95\%$ CP for the true values ${\delta}_{0}=1.0$, ${\delta}_{1}=2.0$ and $\alpha =0.5$ of the indicated quantile level ($\tau $), sample size (n) and parameter (${\delta}_{0},{\delta}_{1},\alpha $) with simulated data.

$\mathit{\tau}$ | n | Bias | RMSE | CP${}_{95\%}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | $\mathit{\alpha}$ | ||

0.10 | 20 | $0.050$ | $-0.007$ | $-0.034$ | $0.202$ | $0.174$ | $0.085$ | $0.887$ | $0.916$ | $0.853$ |

50 | $0.021$ | $-0.003$ | $-0.013$ | $0.123$ | $0.103$ | $0.051$ | $0.927$ | $0.938$ | $0.911$ | |

100 | $0.010$ | $-0.002$ | $-0.006$ | $0.087$ | $0.073$ | $0.036$ | $0.936$ | $0.940$ | $0.932$ | |

200 | $0.005$ | $-0.001$ | $-0.003$ | $0.061$ | $0.051$ | $0.025$ | $0.943$ | $0.944$ | $0.941$ | |

300 | $0.003$ | $-0.000$ | $-0.002$ | $0.050$ | $0.041$ | $0.020$ | $0.948$ | $0.947$ | $0.946$ | |

0.25 | 20 | $0.026$ | $-0.002$ | $-0.034$ | $0.168$ | $0.173$ | $0.086$ | $0.903$ | $0.918$ | $0.854$ |

50 | $0.011$ | $-0.002$ | $-0.013$ | $0.103$ | $0.102$ | $0.051$ | $0.935$ | $0.940$ | $0.911$ | |

100 | $0.005$ | $-0.001$ | $-0.006$ | $0.073$ | $0.072$ | $0.036$ | $0.940$ | $0.941$ | $0.931$ | |

200 | $0.003$ | $-0.000$ | $-0.003$ | $0.051$ | $0.050$ | $0.025$ | $0.944$ | $0.945$ | $0.940$ | |

300 | $0.002$ | $-0.000$ | $-0.002$ | $0.042$ | $0.041$ | $0.020$ | $0.950$ | $0.947$ | $0.946$ | |

0.50 | 20 | $-0.003$ | $0.004$ | $-0.034$ | $0.152$ | $0.172$ | $0.086$ | $0.918$ | $0.918$ | $0.856$ |

50 | $-0.000$ | $0.001$ | $-0.013$ | $0.094$ | $0.102$ | $0.051$ | $0.938$ | $0.941$ | $0.912$ | |

100 | $-0.000$ | $-0.000$ | $-0.006$ | $0.066$ | $0.072$ | $0.036$ | $0.944$ | $0.943$ | $0.932$ | |

200 | $0.000$ | $0.000$ | $-0.003$ | $0.047$ | $0.050$ | $0.025$ | $0.946$ | $0.946$ | $0.940$ | |

300 | $0.000$ | $0.000$ | $-0.002$ | $0.038$ | $0.040$ | $0.020$ | $0.949$ | $0.949$ | $0.947$ | |

0.75 | 20 | $-0.033$ | $0.011$ | $-0.034$ | $0.171$ | $0.174$ | $0.085$ | $0.914$ | $0.920$ | $0.855$ |

50 | $-0.012$ | $0.003$ | $-0.013$ | $0.104$ | $0.103$ | $0.051$ | $0.934$ | $0.944$ | $0.914$ | |

100 | $-0.006$ | $0.001$ | $-0.006$ | $0.073$ | $0.072$ | $0.036$ | $0.944$ | $0.942$ | $0.933$ | |

200 | $-0.003$ | $0.001$ | $-0.003$ | $0.052$ | $0.050$ | $0.025$ | $0.946$ | $0.946$ | $0.940$ | |

300 | $-0.002$ | $0.001$ | $-0.002$ | $0.042$ | $0.041$ | $0.020$ | $0.948$ | $0.949$ | $0.947$ | |

0.90 | 20 | $-0.058$ | $0.018$ | $-0.033$ | $0.207$ | $0.177$ | $0.085$ | $0.904$ | $0.920$ | $0.857$ |

50 | $-0.022$ | $0.006$ | $-0.013$ | $0.125$ | $0.104$ | $0.051$ | $0.930$ | $0.944$ | $0.915$ | |

100 | $-0.011$ | $0.002$ | $-0.006$ | $0.087$ | $0.073$ | $0.036$ | $0.941$ | $0.942$ | $0.935$ | |

200 | $-0.005$ | $0.002$ | $-0.003$ | $0.061$ | $0.051$ | $0.025$ | $0.947$ | $0.947$ | $0.939$ | |

300 | $-0.003$ | $0.001$ | $-0.002$ | $0.049$ | $0.041$ | $0.020$ | $0.950$ | $0.949$ | $0.945$ |

**Table 2.**Empirical bias, RMSE and $95\%$ CP for the true values ${\delta}_{0}=1.0$, ${\delta}_{1}=2.0$ and $\alpha =1.0$ of the indicated quantile level ($\tau $), sample size (n) and parameter (${\delta}_{0},{\delta}_{1},\alpha $) with simulated data.

$\mathit{\tau}$ | n | Bias | RMSE | CP${}_{95\%}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | $\mathit{\alpha}$ | ||

0.10 | 20 | $0.094$ | $-0.007$ | $-0.073$ | $0.367$ | $0.320$ | $0.173$ | $0.884$ | $0.918$ | $0.846$ |

50 | $0.038$ | $-0.005$ | $-0.029$ | $0.222$ | $0.189$ | $0.103$ | $0.925$ | $0.939$ | $0.908$ | |

100 | $0.018$ | $-0.003$ | $-0.014$ | $0.157$ | $0.133$ | $0.072$ | $0.936$ | $0.941$ | $0.930$ | |

200 | $0.009$ | $-0.001$ | $-0.007$ | $0.110$ | $0.092$ | $0.050$ | $0.944$ | $0.946$ | $0.939$ | |

300 | $0.006$ | $-0.000$ | $-0.004$ | $0.089$ | $0.075$ | $0.041$ | $0.948$ | $0.947$ | $0.945$ | |

0.25 | 20 | $0.050$ | $0.001$ | $-0.073$ | $0.312$ | $0.318$ | $0.173$ | $0.901$ | $0.918$ | $0.847$ |

50 | $0.021$ | $-0.002$ | $-0.029$ | $0.190$ | $0.188$ | $0.103$ | $0.934$ | $0.940$ | $0.908$ | |

100 | $0.010$ | $-0.002$ | $-0.014$ | $0.134$ | $0.131$ | $0.072$ | $0.938$ | $0.942$ | $0.930$ | |

200 | $0.005$ | $0.000$ | $-0.007$ | $0.095$ | $0.092$ | $0.050$ | $0.944$ | $0.946$ | $0.939$ | |

300 | $0.003$ | $0.000$ | $-0.004$ | $0.077$ | $0.074$ | $0.041$ | $0.949$ | $0.948$ | $0.945$ | |

0.50 | 20 | $-0.010$ | $0.013$ | $-0.073$ | $0.283$ | $0.320$ | $0.173$ | $0.916$ | $0.922$ | $0.847$ |

50 | $-0.003$ | $0.003$ | $-0.029$ | $0.172$ | $0.188$ | $0.103$ | $0.939$ | $0.942$ | $0.909$ | |

100 | $-0.001$ | $0.000$ | $-0.014$ | $0.121$ | $0.131$ | $0.072$ | $0.945$ | $0.944$ | $0.930$ | |

200 | $-0.000$ | $0.001$ | $-0.007$ | $0.085$ | $0.091$ | $0.050$ | $0.945$ | $0.946$ | $0.940$ | |

300 | $-0.000$ | $0.001$ | $-0.004$ | $0.069$ | $0.074$ | $0.041$ | $0.948$ | $0.948$ | $0.946$ | |

0.75 | 20 | $-0.074$ | $0.029$ | $-0.072$ | $0.325$ | $0.327$ | $0.173$ | $0.915$ | $0.923$ | $0.849$ |

50 | $-0.028$ | $0.009$ | $-0.029$ | $0.194$ | $0.190$ | $0.103$ | $0.936$ | $0.944$ | $0.910$ | |

100 | $-0.013$ | $0.003$ | $-0.014$ | $0.135$ | $0.132$ | $0.072$ | $0.945$ | $0.944$ | $0.931$ | |

200 | $-0.006$ | $0.002$ | $-0.007$ | $0.095$ | $0.092$ | $0.050$ | $0.946$ | $0.947$ | $0.939$ | |

300 | $-0.004$ | $0.002$ | $-0.004$ | $0.077$ | $0.074$ | $0.041$ | $0.950$ | $0.948$ | $0.946$ | |

0.90 | 20 | $-0.122$ | $0.043$ | $-0.071$ | $0.390$ | $0.335$ | $0.171$ | $0.908$ | $0.924$ | $0.850$ |

50 | $-0.047$ | $0.014$ | $-0.028$ | $0.228$ | $0.192$ | $0.102$ | $0.932$ | $0.946$ | $0.912$ | |

100 | $-0.023$ | $0.005$ | $-0.014$ | $0.158$ | $0.133$ | $0.071$ | $0.943$ | $0.944$ | $0.933$ | |

200 | $-0.010$ | $0.003$ | $-0.007$ | $0.111$ | $0.093$ | $0.050$ | $0.947$ | $0.947$ | $0.939$ | |

300 | $-0.007$ | $0.002$ | $-0.004$ | $0.089$ | $0.075$ | $0.040$ | $0.949$ | $0.950$ | $0.945$ |

**Table 3.**Empirical bias, RMSE and $95\%$ CP for the true values ${\delta}_{0}=1.0$, ${\delta}_{1}=2.0$ and $\alpha =2.0$ of the indicated quantile level ($\tau $), sample size (n) and parameter (${\delta}_{0},{\delta}_{1},\alpha $) with simulated data.

$\mathit{\tau}$ | n | Bias | RMSE | CP${}_{95\%}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | $\mathit{\alpha}$ | ||

0.10 | 20 | $0.152$ | $-0.004$ | $-0.162$ | $0.566$ | $0.511$ | $0.358$ | $0.887$ | $0.921$ | $0.831$ |

50 | $0.061$ | $-0.005$ | $-0.064$ | $0.333$ | $0.292$ | $0.209$ | $0.928$ | $0.941$ | $0.901$ | |

100 | $0.029$ | $-0.004$ | $-0.031$ | $0.233$ | $0.201$ | $0.145$ | $0.938$ | $0.941$ | $0.928$ | |

200 | $0.014$ | $-0.001$ | $-0.015$ | $0.163$ | $0.139$ | $0.101$ | $0.945$ | $0.947$ | $0.938$ | |

300 | $0.009$ | $0.000$ | $-0.009$ | $0.132$ | $0.112$ | $0.081$ | $0.950$ | $0.947$ | $0.944$ | |

0.25 | 20 | $0.095$ | $0.006$ | $-0.162$ | $0.502$ | $0.512$ | $0.358$ | $0.897$ | $0.922$ | $0.833$ |

50 | $0.039$ | $-0.002$ | $-0.064$ | $0.297$ | $0.291$ | $0.210$ | $0.930$ | $0.942$ | $0.901$ | |

100 | $0.019$ | $-0.002$ | $-0.031$ | $0.208$ | $0.200$ | $0.145$ | $0.941$ | $0.943$ | $0.928$ | |

200 | $0.010$ | $0.000$ | $-0.015$ | $0.146$ | $0.138$ | $0.101$ | $0.946$ | $0.947$ | $0.938$ | |

300 | $0.006$ | $0.001$ | $-0.009$ | $0.118$ | $0.111$ | $0.082$ | $0.948$ | $0.946$ | $0.944$ | |

0.50 | 20 | $-0.026$ | $0.032$ | $-0.160$ | $0.450$ | $0.522$ | $0.358$ | $0.917$ | $0.924$ | $0.832$ |

50 | $-0.008$ | $0.007$ | $-0.063$ | $0.262$ | $0.292$ | $0.210$ | $0.939$ | $0.943$ | $0.903$ | |

100 | $-0.004$ | $0.002$ | $-0.030$ | $0.181$ | $0.199$ | $0.145$ | $0.945$ | $0.944$ | $0.929$ | |

200 | $-0.001$ | $0.002$ | $-0.014$ | $0.127$ | $0.138$ | $0.101$ | $0.946$ | $0.946$ | $0.938$ | |

300 | $-0.001$ | $0.002$ | $-0.009$ | $0.102$ | $0.111$ | $0.082$ | $0.949$ | $0.948$ | $0.945$ | |

0.75 | 20 | $-0.157$ | $0.069$ | $-0.158$ | $0.558$ | $0.549$ | $0.356$ | $0.922$ | $0.928$ | $0.833$ |

50 | $-0.058$ | $0.020$ | $-0.062$ | $0.311$ | $0.299$ | $0.208$ | $0.941$ | $0.945$ | $0.905$ | |

100 | $-0.028$ | $0.008$ | $-0.030$ | $0.212$ | $0.202$ | $0.144$ | $0.947$ | $0.945$ | $0.929$ | |

200 | $-0.013$ | $0.005$ | $-0.014$ | $0.148$ | $0.139$ | $0.101$ | $0.946$ | $0.946$ | $0.938$ | |

300 | $-0.009$ | $0.004$ | $-0.009$ | $0.119$ | $0.111$ | $0.081$ | $0.949$ | $0.949$ | $0.946$ | |

0.90 | 20 | $-0.227$ | $0.087$ | $-0.160$ | $0.653$ | $0.569$ | $0.354$ | $0.918$ | $0.930$ | $0.833$ |

50 | $-0.083$ | $0.024$ | $-0.065$ | $0.353$ | $0.303$ | $0.208$ | $0.938$ | $0.947$ | $0.904$ | |

100 | $-0.040$ | $0.007$ | $-0.033$ | $0.238$ | $0.203$ | $0.144$ | $0.945$ | $0.946$ | $0.928$ | |

200 | $-0.020$ | $0.003$ | $-0.018$ | $0.166$ | $0.140$ | $0.101$ | $0.947$ | $0.946$ | $0.935$ | |

300 | $-0.014$ | $0.001$ | $-0.013$ | $0.133$ | $0.112$ | $0.081$ | $0.948$ | $0.950$ | $0.942$ |

**Table 4.**Empirical bias, RMSE and $95\%$ CP for the true values ${\delta}_{0}=1.0$, ${\delta}_{1}=1.0$, ${\delta}_{2}=2.0$ and $\alpha =0.5$ of the indicated quantile level ($\tau $), sample size (n) and parameter (${\delta}_{0},{\delta}_{1},\alpha $) with simulated data.

$\mathit{\tau}$ | n | Bias | RMSE | CP${}_{95\%}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | $\mathit{\alpha}$ | ||

0.10 | 20 | $0.066$ | $-0.006$ | $-0.010$ | $-0.046$ | $0.213$ | $0.160$ | $0.176$ | $0.091$ | $0.876$ | $0.916$ | $0.912$ | $0.813$ |

50 | $0.027$ | $-0.002$ | $-0.005$ | $-0.019$ | $0.128$ | $0.095$ | $0.103$ | $0.052$ | $0.920$ | $0.930$ | $0.938$ | $0.901$ | |

100 | $0.013$ | $-0.001$ | $-0.002$ | $-0.009$ | $0.090$ | $0.065$ | $0.073$ | $0.036$ | $0.933$ | $0.939$ | $0.941$ | $0.924$ | |

200 | $0.006$ | $-0.000$ | $-0.001$ | $-0.005$ | $0.063$ | $0.045$ | $0.051$ | $0.025$ | $0.939$ | $0.946$ | $0.945$ | $0.934$ | |

300 | $0.004$ | $-0.001$ | $-0.000$ | $-0.003$ | $0.052$ | $0.036$ | $0.041$ | $0.021$ | $0.941$ | $0.948$ | $0.945$ | $0.940$ | |

0.25 | 20 | $0.033$ | $-0.003$ | $-0.003$ | $-0.047$ | $0.176$ | $0.160$ | $0.175$ | $0.091$ | $0.898$ | $0.916$ | $0.915$ | $0.812$ |

50 | $0.014$ | $-0.001$ | $-0.002$ | $-0.019$ | $0.107$ | $0.095$ | $0.102$ | $0.053$ | $0.928$ | $0.930$ | $0.940$ | $0.899$ | |

100 | $0.007$ | $-0.000$ | $-0.001$ | $-0.009$ | $0.075$ | $0.065$ | $0.072$ | $0.037$ | $0.940$ | $0.939$ | $0.942$ | $0.924$ | |

200 | $0.003$ | $-0.000$ | $-0.000$ | $-0.005$ | $0.053$ | $0.045$ | $0.050$ | $0.026$ | $0.941$ | $0.946$ | $0.946$ | $0.934$ | |

300 | $0.002$ | $-0.000$ | $0.000$ | $-0.003$ | $0.043$ | $0.036$ | $0.041$ | $0.021$ | $0.941$ | $0.947$ | $0.944$ | $0.939$ | |

0.50 | 20 | $-0.008$ | $0.001$ | $0.005$ | $-0.047$ | $0.158$ | $0.161$ | $0.175$ | $0.092$ | $0.914$ | $0.917$ | $0.919$ | $0.813$ |

50 | $-0.003$ | $0.001$ | $0.001$ | $-0.019$ | $0.097$ | $0.095$ | $0.102$ | $0.053$ | $0.937$ | $0.930$ | $0.941$ | $0.895$ | |

100 | $-0.001$ | $0.000$ | $0.001$ | $-0.009$ | $0.069$ | $0.065$ | $0.072$ | $0.037$ | $0.940$ | $0.938$ | $0.943$ | $0.922$ | |

200 | $-0.001$ | $0.000$ | $0.001$ | $-0.005$ | $0.048$ | $0.045$ | $0.050$ | $0.026$ | $0.945$ | $0.947$ | $0.947$ | $0.931$ | |

300 | $-0.001$ | $-0.000$ | $0.001$ | $-0.003$ | $0.039$ | $0.036$ | $0.041$ | $0.021$ | $0.944$ | $0.947$ | $0.944$ | $0.939$ | |

0.75 | 20 | $-0.049$ | $0.006$ | $0.015$ | $-0.046$ | $0.180$ | $0.162$ | $0.177$ | $0.091$ | $0.899$ | $0.917$ | $0.920$ | $0.813$ |

50 | $-0.020$ | $0.002$ | $0.005$ | $-0.019$ | $0.109$ | $0.095$ | $0.103$ | $0.053$ | $0.928$ | $0.931$ | $0.942$ | $0.894$ | |

100 | $-0.010$ | $0.001$ | $0.002$ | $-0.009$ | $0.076$ | $0.065$ | $0.072$ | $0.037$ | $0.937$ | $0.938$ | $0.942$ | $0.920$ | |

200 | $-0.005$ | $0.001$ | $0.001$ | $-0.005$ | $0.053$ | $0.045$ | $0.050$ | $0.026$ | $0.941$ | $0.948$ | $0.946$ | $0.932$ | |

300 | $-0.004$ | $0.000$ | $0.001$ | $-0.003$ | $0.043$ | $0.036$ | $0.041$ | $0.021$ | $0.945$ | $0.947$ | $0.945$ | $0.938$ | |

0.90 | 20 | $-0.085$ | $0.010$ | $0.024$ | $-0.046$ | $0.222$ | $0.164$ | $0.181$ | $0.090$ | $0.879$ | $0.917$ | $0.919$ | $0.815$ |

50 | $-0.034$ | $0.004$ | $0.008$ | $-0.019$ | $0.131$ | $0.096$ | $0.105$ | $0.053$ | $0.921$ | $0.931$ | $0.940$ | $0.894$ | |

100 | $-0.017$ | $0.002$ | $0.004$ | $-0.009$ | $0.091$ | $0.065$ | $0.073$ | $0.036$ | $0.934$ | $0.937$ | $0.943$ | $0.920$ | |

200 | $-0.009$ | $0.001$ | $0.002$ | $-0.005$ | $0.064$ | $0.045$ | $0.051$ | $0.025$ | $0.940$ | $0.948$ | $0.947$ | $0.933$ | |

300 | $-0.006$ | $0.000$ | $0.002$ | $-0.003$ | $0.051$ | $0.036$ | $0.041$ | $0.021$ | $0.945$ | $0.948$ | $0.945$ | $0.937$ |

**Table 5.**Empirical bias, RMSE and $95\%$ CP for the true values: ${\delta}_{0}=1.0$, ${\delta}_{1}=1.0$, ${\delta}_{2}=2.0$ and $\alpha =1.0$ of the indicated quantile level ($\tau $), sample size (n) and parameter (${\delta}_{0},{\delta}_{1},\alpha $) with simulated data.

$\mathit{\tau}$ | n | Bias | RMSE | CP${}_{95\%}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | $\mathit{\alpha}$ | ||

0.10 | 20 | $0.124$ | $-0.009$ | $-0.013$ | $-0.101$ | $0.388$ | $0.297$ | $0.326$ | $0.185$ | $0.871$ | $0.916$ | $0.914$ | $0.799$ |

50 | $0.050$ | $-0.003$ | $-0.007$ | $-0.041$ | $0.231$ | $0.175$ | $0.190$ | $0.106$ | $0.919$ | $0.932$ | $0.942$ | $0.894$ | |

100 | $0.025$ | $-0.001$ | $-0.003$ | $-0.020$ | $0.162$ | $0.119$ | $0.133$ | $0.073$ | $0.933$ | $0.939$ | $0.941$ | $0.922$ | |

200 | $0.012$ | $-0.001$ | $-0.001$ | $-0.010$ | $0.113$ | $0.082$ | $0.092$ | $0.051$ | $0.940$ | $0.949$ | $0.943$ | $0.933$ | |

300 | $0.008$ | $-0.001$ | $-0.000$ | $-0.007$ | $0.093$ | $0.066$ | $0.075$ | $0.041$ | $0.943$ | $0.948$ | $0.945$ | $0.939$ | |

0.25 | 20 | $0.063$ | $-0.004$ | $-0.001$ | $-0.101$ | $0.327$ | $0.298$ | $0.325$ | $0.186$ | $0.895$ | $0.919$ | $0.916$ | $0.799$ |

50 | $0.026$ | $-0.001$ | $-0.002$ | $-0.041$ | $0.198$ | $0.175$ | $0.189$ | $0.106$ | $0.927$ | $0.933$ | $0.942$ | $0.893$ | |

100 | $0.013$ | $-0.000$ | $-0.001$ | $-0.020$ | $0.139$ | $0.119$ | $0.132$ | $0.073$ | $0.936$ | $0.939$ | $0.944$ | $0.921$ | |

200 | $0.006$ | $-0.000$ | $-0.000$ | $-0.010$ | $0.097$ | $0.082$ | $0.092$ | $0.051$ | $0.940$ | $0.949$ | $0.945$ | $0.932$ | |

300 | $0.004$ | $-0.000$ | $0.000$ | $-0.007$ | $0.080$ | $0.066$ | $0.074$ | $0.042$ | $0.942$ | $0.947$ | $0.945$ | $0.938$ | |

0.50 | 20 | $-0.022$ | $0.005$ | $0.017$ | $-0.101$ | $0.296$ | $0.301$ | $0.328$ | $0.187$ | $0.915$ | $0.919$ | $0.921$ | $0.801$ |

50 | $-0.008$ | $0.002$ | $0.005$ | $-0.041$ | $0.179$ | $0.175$ | $0.189$ | $0.107$ | $0.937$ | $0.933$ | $0.943$ | $0.891$ | |

100 | $-0.004$ | $0.001$ | $0.002$ | $-0.020$ | $0.126$ | $0.119$ | $0.131$ | $0.074$ | $0.940$ | $0.939$ | $0.944$ | $0.920$ | |

200 | $-0.003$ | $0.001$ | $0.002$ | $-0.010$ | $0.088$ | $0.082$ | $0.091$ | $0.052$ | $0.944$ | $0.949$ | $0.946$ | $0.930$ | |

300 | $-0.002$ | $0.000$ | $0.001$ | $-0.007$ | $0.072$ | $0.066$ | $0.074$ | $0.042$ | $0.946$ | $0.948$ | $0.945$ | $0.938$ | |

0.75 | 20 | $-0.110$ | $0.016$ | $0.038$ | $-0.100$ | $0.349$ | $0.306$ | $0.337$ | $0.186$ | $0.902$ | $0.920$ | $0.921$ | $0.801$ |

50 | $-0.044$ | $0.006$ | $0.013$ | $-0.041$ | $0.204$ | $0.177$ | $0.191$ | $0.107$ | $0.931$ | $0.933$ | $0.944$ | $0.889$ | |

100 | $-0.021$ | $0.003$ | $0.006$ | $-0.020$ | $0.142$ | $0.119$ | $0.132$ | $0.074$ | $0.934$ | $0.939$ | $0.944$ | $0.918$ | |

200 | $-0.012$ | $0.002$ | $0.003$ | $-0.010$ | $0.099$ | $0.082$ | $0.092$ | $0.051$ | $0.943$ | $0.949$ | $0.947$ | $0.930$ | |

300 | $-0.008$ | $0.001$ | $0.003$ | $-0.007$ | $0.080$ | $0.066$ | $0.074$ | $0.042$ | $0.946$ | $0.949$ | $0.945$ | $0.937$ | |

0.90 | 20 | $-0.177$ | $0.025$ | $0.056$ | $-0.098$ | $0.425$ | $0.311$ | $0.347$ | $0.184$ | $0.885$ | $0.922$ | $0.921$ | $0.801$ |

50 | $-0.070$ | $0.010$ | $0.019$ | $-0.040$ | $0.242$ | $0.178$ | $0.195$ | $0.106$ | $0.924$ | $0.934$ | $0.942$ | $0.889$ | |

100 | $-0.034$ | $0.005$ | $0.009$ | $-0.020$ | $0.166$ | $0.120$ | $0.134$ | $0.073$ | $0.933$ | $0.939$ | $0.945$ | $0.918$ | |

200 | $-0.018$ | $0.002$ | $0.005$ | $-0.010$ | $0.115$ | $0.082$ | $0.093$ | $0.051$ | $0.942$ | $0.949$ | $0.948$ | $0.929$ | |

300 | $-0.012$ | $0.001$ | $0.004$ | $-0.007$ | $0.093$ | $0.066$ | $0.075$ | $0.041$ | $0.944$ | $0.949$ | $0.945$ | $0.937$ |

**Table 6.**Empirical bias, RMSE and $95\%$ CP for the true values ${\delta}_{0}=1.0$, ${\delta}_{1}=1.0$, ${\delta}_{2}=2.0$ and $\alpha =2.0$ of the indicated quantile level ($\tau $), sample size (n) and parameter (${\delta}_{0},{\delta}_{1},\alpha $) with simulated data.

$\mathit{\tau}$ | n | Bias | RMSE | CP${}_{95\%}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | $\mathit{\alpha}$ | ${\mathit{\delta}}_{\mathbf{0}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | $\mathit{\alpha}$ | ||

0.10 | 20 | $0.203$ | $-0.012$ | $-0.009$ | $-0.225$ | $0.613$ | $0.489$ | $0.534$ | $0.391$ | $0.867$ | $0.917$ | $0.913$ | $0.775$ |

50 | $0.080$ | $-0.003$ | $-0.008$ | $-0.090$ | $0.352$ | $0.275$ | $0.297$ | $0.217$ | $0.919$ | $0.937$ | $0.941$ | $0.884$ | |

100 | $0.040$ | $-0.001$ | $-0.004$ | $-0.044$ | $0.243$ | $0.183$ | $0.203$ | $0.148$ | $0.929$ | $0.942$ | $0.944$ | $0.916$ | |

200 | $0.019$ | $-0.000$ | $-0.001$ | $-0.022$ | $0.169$ | $0.123$ | $0.139$ | $0.103$ | $0.939$ | $0.951$ | $0.944$ | $0.929$ | |

300 | $0.012$ | $-0.001$ | $-0.000$ | $-0.015$ | $0.137$ | $0.100$ | $0.112$ | $0.083$ | $0.942$ | $0.950$ | $0.946$ | $0.937$ | |

0.25 | 20 | $0.123$ | $-0.005$ | $0.006$ | $-0.224$ | $0.540$ | $0.493$ | $0.537$ | $0.392$ | $0.887$ | $0.917$ | $0.916$ | $0.776$ |

50 | $0.050$ | $-0.000$ | $-0.002$ | $-0.090$ | $0.314$ | $0.276$ | $0.296$ | $0.218$ | $0.926$ | $0.938$ | $0.942$ | $0.884$ | |

100 | $0.025$ | $0.000$ | $-0.002$ | $-0.044$ | $0.218$ | $0.182$ | $0.202$ | $0.149$ | $0.933$ | $0.942$ | $0.945$ | $0.917$ | |

200 | $0.011$ | $0.000$ | $0.000$ | $-0.022$ | $0.151$ | $0.123$ | $0.139$ | $0.103$ | $0.942$ | $0.951$ | $0.945$ | $0.929$ | |

300 | $0.007$ | $-0.000$ | $0.001$ | $-0.015$ | $0.123$ | $0.100$ | $0.111$ | $0.084$ | $0.942$ | $0.950$ | $0.946$ | $0.937$ | |

0.50 | 20 | $-0.049$ | $0.013$ | $0.043$ | $-0.223$ | $0.488$ | $0.506$ | $0.552$ | $0.392$ | $0.919$ | $0.919$ | $0.921$ | $0.779$ |

50 | $-0.017$ | $0.006$ | $0.012$ | $-0.090$ | $0.277$ | $0.278$ | $0.298$ | $0.220$ | $0.938$ | $0.939$ | $0.944$ | $0.882$ | |

100 | $-0.008$ | $0.004$ | $0.005$ | $-0.045$ | $0.190$ | $0.183$ | $0.201$ | $0.150$ | $0.941$ | $0.942$ | $0.947$ | $0.916$ | |

200 | $-0.005$ | $0.002$ | $0.003$ | $-0.023$ | $0.132$ | $0.123$ | $0.138$ | $0.104$ | $0.945$ | $0.951$ | $0.947$ | $0.927$ | |

300 | $-0.004$ | $0.001$ | $0.003$ | $-0.015$ | $0.107$ | $0.100$ | $0.111$ | $0.084$ | $0.946$ | $0.951$ | $0.947$ | $0.936$ | |

0.75 | 20 | $-0.234$ | $0.038$ | $0.093$ | $-0.220$ | $0.624$ | $0.528$ | $0.585$ | $0.389$ | $0.909$ | $0.923$ | $0.923$ | $0.779$ |

50 | $-0.089$ | $0.015$ | $0.029$ | $-0.089$ | $0.335$ | $0.282$ | $0.306$ | $0.219$ | $0.933$ | $0.940$ | $0.945$ | $0.879$ | |

100 | $-0.043$ | $0.007$ | $0.012$ | $-0.045$ | $0.225$ | $0.184$ | $0.203$ | $0.149$ | $0.936$ | $0.942$ | $0.947$ | $0.913$ | |

200 | $-0.023$ | $0.004$ | $0.007$ | $-0.023$ | $0.155$ | $0.124$ | $0.139$ | $0.103$ | $0.943$ | $0.951$ | $0.947$ | $0.927$ | |

300 | $-0.016$ | $0.002$ | $0.005$ | $-0.015$ | $0.124$ | $0.100$ | $0.112$ | $0.083$ | $0.944$ | $0.950$ | $0.948$ | $0.936$ | |

0.90 | 20 | $-0.330$ | $0.051$ | $0.119$ | $-0.222$ | $0.739$ | $0.541$ | $0.607$ | $0.387$ | $0.902$ | $0.925$ | $0.927$ | $0.776$ |

50 | $-0.126$ | $0.018$ | $0.035$ | $-0.094$ | $0.384$ | $0.285$ | $0.311$ | $0.219$ | $0.928$ | $0.940$ | $0.946$ | $0.876$ | |

100 | $-0.063$ | $0.008$ | $0.013$ | $-0.050$ | $0.255$ | $0.186$ | $0.205$ | $0.149$ | $0.935$ | $0.941$ | $0.947$ | $0.910$ | |

200 | $-0.033$ | $0.003$ | $0.005$ | $-0.028$ | $0.174$ | $0.124$ | $0.140$ | $0.104$ | $0.940$ | $0.951$ | $0.948$ | $0.923$ | |

300 | $-0.023$ | $0.001$ | $0.003$ | $-0.020$ | $0.139$ | $0.101$ | $0.113$ | $0.084$ | $0.942$ | $0.947$ | $0.950$ | $0.932$ |

**Table 7.**ML estimates and SEs of the indicated model, parameter and quantile level ($\tau $) with vote data.

Model | Parameter | Estimate | SE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\tau}$ Level | $\mathit{\tau}$ Level | ||||||||||

0.10 | 0.25 | 0.50 | 0.75 | 0.90 | 0.10 | 0.25 | 0.50 | 0.75 | 0.90 | ||

KUMA | $\alpha $ | $12.204$ | $12.209$ | $12.215$ | $12.222$ | $12.228$ | $1.152$ | $1.152$ | $1.152$ | $1.151$ | $1.151$ |

${\delta}_{0}$ | $1.717$ | $2.153$ | $2.649$ | $3.158$ | $3.629$ | $0.462$ | $0.529$ | $0.612$ | $0.702$ | $0.786$ | |

${\delta}_{1}$ | $-2.081$ | $-2.425$ | $-2.862$ | $-3.352$ | $-3.835$ | $0.764$ | $0.877$ | $1.015$ | $1.161$ | $1.296$ | |

LEEG | $\alpha $ | $18.176$ | $18.183$ | $18.190$ | $18.198$ | $18.207$ | $1.825$ | $1.825$ | $1.825$ | $1.826$ | $1.826$ |

${\delta}_{0}$ | $2.155$ | $2.506$ | $2.937$ | $3.481$ | $4.197$ | $0.415$ | $0.450$ | $0.496$ | $0.562$ | $0.660$ | |

${\delta}_{1}$ | $-2.708$ | $-3.005$ | $-3.395$ | $-3.924$ | $-4.662$ | $0.677$ | $0.738$ | $0.816$ | $0.919$ | $1.062$ | |

UBS | $\alpha $ | $0.279$ | $0.279$ | $0.279$ | $0.279$ | $0.279$ | $0.023$ | $0.023$ | $0.023$ | $0.023$ | $0.023$ |

${\delta}_{0}$ | $2.457$ | $2.596$ | $2.760$ | $2.925$ | $3.077$ | $0.670$ | $0.645$ | $0.623$ | $0.605$ | $0.592$ | |

${\delta}_{1}$ | $-3.281$ | $-3.168$ | $-3.066$ | $-2.978$ | $-2.916$ | $1.123$ | $1.081$ | $1.043$ | $1.012$ | $0.989$ | |

UBURR | $\alpha $ | $40.007$ | $40.004$ | $40.085$ | $40.663$ | $40.661$ | $11.258$ | $11.249$ | $11.280$ | $11.539$ | $11.538$ |

${\delta}_{0}$ | $1.907$ | $2.294$ | $2.784$ | $3.398$ | $4.068$ | $0.423$ | $0.484$ | $0.570$ | $0.671$ | $0.788$ | |

${\delta}_{1}$ | $-2.368$ | $-2.733$ | $-3.184$ | $-3.763$ | $-4.390$ | $0.692$ | $0.797$ | $0.938$ | $1.102$ | $1.289$ | |

UCHEN | $\alpha $ | $3.712$ | $3.712$ | $3.712$ | $3.712$ | $3.712$ | $0.335$ | $0.335$ | $0.335$ | $0.334$ | $0.334$ |

${\delta}_{0}$ | $2.246$ | $2.392$ | $2.578$ | $2.803$ | $3.057$ | $0.665$ | $0.661$ | $0.650$ | $0.634$ | $0.619$ | |

${\delta}_{1}$ | $-2.939$ | $-2.908$ | $-2.848$ | $-2.766$ | $-2.683$ | $1.113$ | $1.106$ | $1.087$ | $1.057$ | $1.026$ | |

UHN | ${\delta}_{0}$ | $0.742$ | $1.672$ | $2.422$ | $2.956$ | $3.314$ | $1.408$ | $1.408$ | $1.408$ | $1.408$ | $1.408$ |

${\delta}_{1}$ | $-3.000$ | $-3.000$ | $-3.000$ | $-3.000$ | $-3.000$ | $2.353$ | $2.353$ | $2.353$ | $2.353$ | $2.353$ | |

ULOG | $\alpha $ | $5.304$ | $5.304$ | $5.304$ | $5.305$ | $5.304$ | $0.512$ | $0.512$ | $0.512$ | $0.512$ | $0.512$ |

${\delta}_{0}$ | $2.506$ | $2.713$ | $2.921$ | $3.127$ | $3.335$ | $0.566$ | $0.564$ | $0.563$ | $0.563$ | $0.564$ | |

${\delta}_{1}$ | $-3.338$ | $-3.338$ | $-3.338$ | $-3.338$ | $-3.338$ | $0.939$ | $0.939$ | $0.939$ | $0.939$ | $0.939$ | |

>UWEI | $\alpha $ | $3.768$ | $3.768$ | $3.767$ | $3.767$ | $3.767$ | $0.324$ | $0.324$ | $0.324$ | $0.324$ | $0.324$ |

${\delta}_{0}$ | $2.302$ | $2.414$ | $2.573$ | $2.781$ | $3.027$ | $0.697$ | $0.676$ | $0.652$ | $0.629$ | $0.610$ | |

${\delta}_{1}$ | $-3.032$ | $-2.940$ | $-2.835$ | $-2.729$ | $-2.637$ | $1.169$ | $1.132$ | $1.091$ | $1.049$ | $1.013$ |

**Table 8.**Values of the Akaike information criterion for the indicated model and quantile level ($\tau $) with vote data.

Model | $\mathit{\tau}$ Level | ||||
---|---|---|---|---|---|

0.10 | 0.25 | 0.50 | 0.75 | 0.90 | |

KUMA | $-186.6585$ | $-186.7418$ | $-186.8450$ | $-186.9566$ | $-187.0622$ |

LEEG | $-181.1746$ | $-181.2213$ | $-181.2757$ | $-181.3365$ | $-181.3965$ |

UBS | $-188.7764$ | $-188.7825$ | $-188.7883$ | $-188.7932$ | $-188.7969$ |

UBUR | $-180.7540$ | $-180.7524$ | $-180.7513$ | $-180.7531$ | $-180.7528$ |

UCHE | $-178.9413$ | $-178.9489$ | $-178.9513$ | $-178.9480$ | $-178.9416$ |

UHN | $-112.5021$ | $-112.5021$ | $-112.5021$ | $-112.5021$ | $-112.5021$ |

ULOG | $-186.1657$ | $-186.1657$ | $-186.1657$ | $-186.1657$ | $-186.1657$ |

UWEI | $-179.3758$ | $-179.3678$ | $-179.3581$ | $-179.3479$ | $-179.3385$ |

**Table 9.**ML estimates and SEs of the indicated model, parameter and quantile level ($\tau $) with arm data.

Model | Parameter | Estimate | SE | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\tau}$ Level | $\mathit{\tau}$ Level | ||||||||||

0.10 | 0.25 | 0.50 | 0.75 | 0.90 | 0.10 | 0.25 | 0.50 | 0.75 | 0.90 | ||

KUMA | $\alpha $ | $4.705$ | $4.715$ | $4.724$ | $4.731$ | $4.736$ | $0.210$ | $0.210$ | $0.210$ | $0.210$ | $0.210$ |

${\delta}_{0}$ | $-3.071$ | $-2.894$ | $-2.754$ | $-2.658$ | $-2.599$ | $0.135$ | $0.140$ | $0.147$ | $0.156$ | $0.163$ | |

${\delta}_{1}$ | $0.004$ | $0.004$ | $0.004$ | $0.004$ | $0.005$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | |

${\delta}_{2}$ | $7.291$ | $7.721$ | $8.226$ | $8.743$ | $9.210$ | $0.551$ | $0.577$ | $0.609$ | $0.644$ | $0.676$ | |

${\delta}_{3}$ | $-0.730$ | $-0.771$ | $-0.820$ | $-0.870$ | $-0.915$ | $0.033$ | $0.035$ | $0.037$ | $0.039$ | $0.041$ | |

${\delta}_{4}$ | $-0.072$ | $-0.074$ | $-0.076$ | $-0.078$ | $-0.078$ | $0.047$ | $0.050$ | $0.054$ | $0.057$ | $0.060$ | |

${\delta}_{5}$ | $-0.196$ | $-0.206$ | $-0.216$ | $-0.227$ | $-0.236$ | $0.047$ | $0.050$ | $0.053$ | $0.056$ | $0.060$ | |

LEEG | $\alpha $ | $7.587$ | $7.609$ | $7.629$ | $7.648$ | $7.671$ | $0.375$ | $0.375$ | $0.375$ | $0.375$ | $0.374$ |

${\delta}_{0}$ | $-3.159$ | $-3.056$ | $-2.968$ | $-2.897$ | $-2.850$ | $0.142$ | $0.147$ | $0.154$ | $0.163$ | $0.175$ | |

${\delta}_{1}$ | $0.004$ | $0.005$ | $0.005$ | $0.005$ | $0.006$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | |

${\delta}_{2}$ | $8.331$ | $8.746$ | $9.269$ | $9.934$ | $10.785$ | $0.650$ | $0.676$ | $0.711$ | $0.757$ | $0.817$ | |

${\delta}_{3}$ | $-0.870$ | $-0.909$ | $-0.957$ | $-1.020$ | $-1.100$ | $0.035$ | $0.037$ | $0.039$ | $0.041$ | $0.045$ | |

${\delta}_{4}$ | $-0.114$ | $-0.122$ | $-0.131$ | $-0.143$ | $-0.159$ | $0.051$ | $0.053$ | $0.056$ | $0.060$ | $0.065$ | |

${\delta}_{5}$ | $-0.234$ | $-0.244$ | $-0.257$ | $-0.272$ | $-0.292$ | $0.047$ | $0.050$ | $0.052$ | $0.055$ | $0.060$ | |

UBS | $\alpha $ | $0.160$ | $0.160$ | $0.160$ | $0.159$ | $0.159$ | $0.007$ | $0.007$ | $0.007$ | $0.007$ | $0.007$ |

${\delta}_{0}$ | $-3.394$ | $-3.105$ | $-2.801$ | $-2.522$ | $-2.289$ | $0.160$ | $0.150$ | $0.141$ | $0.134$ | $0.129$ | |

${\delta}_{1}$ | $0.005$ | $0.005$ | $0.004$ | $0.004$ | $0.004$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | |

${\delta}_{2}$ | $9.525$ | $9.045$ | $8.581$ | $8.161$ | $7.820$ | $0.705$ | $0.666$ | $0.629$ | $0.596$ | $0.572$ | |

${\delta}_{3}$ | $-1.001$ | $-0.947$ | $-0.896$ | $-0.850$ | $-0.813$ | $0.040$ | $0.038$ | $0.035$ | $0.033$ | $0.032$ | |

${\delta}_{4}$ | $-0.127$ | $-0.122$ | $-0.116$ | $-0.111$ | $-0.108$ | $0.053$ | $0.051$ | $0.048$ | $0.046$ | $0.044$ | |

${\delta}_{5}$ | $-0.271$ | $-0.256$ | $-0.244$ | $-0.232$ | $-0.223$ | $0.052$ | $0.049$ | $0.047$ | $0.044$ | $0.042$ | |

UBUR | $\alpha $ | $0.453$ | $0.457$ | $0.459$ | $0.461$ | $0.463$ | $0.043$ | $0.043$ | $0.043$ | $0.043$ | $0.043$ |

${\delta}_{0}$ | $-1.485$ | $-1.109$ | $-0.782$ | $-0.514$ | $-0.283$ | $0.496$ | $0.257$ | $0.098$ | $0.030$ | $0.109$ | |

${\delta}_{1}$ | $0.002$ | $0.001$ | $0.000$ | $-0.000$ | $-0.000$ | $0.006$ | $0.003$ | $0.001$ | $0.000$ | $0.001$ | |

${\delta}_{2}$ | $-0.302$ | $-0.003$ | $0.045$ | $-0.008$ | $-0.095$ | $2.675$ | $1.346$ | $0.507$ | $0.056$ | $0.482$ | |

${\delta}_{3}$ | $-1.035$ | $-0.495$ | $-0.182$ | $0.019$ | $0.168$ | $0.225$ | $0.102$ | $0.037$ | $0.019$ | $0.044$ | |

${\delta}_{4}$ | $0.114$ | $0.057$ | $0.021$ | $-0.002$ | $-0.020$ | $0.186$ | $0.095$ | $0.036$ | $0.004$ | $0.034$ | |

${\delta}_{5}$ | $-0.122$ | $-0.072$ | $-0.030$ | $0.003$ | $0.030$ | $0.207$ | $0.105$ | $0.040$ | $0.005$ | $0.038$ | |

UCHEN | $\alpha $ | $1.799$ | $1.805$ | $1.809$ | $1.805$ | $1.783$ | $0.039$ | $0.039$ | $0.039$ | $0.040$ | $0.041$ |

${\delta}_{0}$ | $-3.572$ | $-3.503$ | $-3.396$ | $-3.214$ | $-2.872$ | $0.189$ | $0.203$ | $0.223$ | $0.252$ | $0.293$ | |

${\delta}_{1}$ | $0.005$ | $0.006$ | $0.007$ | $0.007$ | $0.008$ | $0.002$ | $0.002$ | $0.002$ | $0.002$ | $0.002$ | |

${\delta}_{2}$ | $9.700$ | $10.393$ | $11.374$ | $12.547$ | $13.522$ | $0.825$ | $0.880$ | $0.959$ | $1.068$ | $1.218$ | |

${\delta}_{3}$ | $-0.962$ | $-1.033$ | $-1.138$ | $-1.265$ | $-1.371$ | $0.048$ | $0.052$ | $0.056$ | $0.064$ | $0.073$ | |

${\delta}_{4}$ | $-0.120$ | $-0.128$ | $-0.138$ | $-0.144$ | $-0.139$ | $0.075$ | $0.080$ | $0.087$ | $0.096$ | $0.103$ | |

${\delta}_{5}$ | $-0.329$ | $-0.353$ | $-0.387$ | $-0.428$ | $-0.460$ | $0.073$ | $0.078$ | $0.085$ | $0.094$ | $0.104$ | |

UHN | ${\delta}_{0}$ | $-4.854$ | $-3.923$ | $-3.173$ | $-2.639$ | $-2.282$ | $0.350$ | $0.350$ | $0.350$ | $0.350$ | $0.350$ |

${\delta}_{1}$ | $0.004$ | $0.004$ | $0.004$ | $0.004$ | $0.004$ | $0.003$ | $0.003$ | $0.003$ | $0.003$ | $0.003$ | |

${\delta}_{2}$ | $8.754$ | $8.754$ | $8.755$ | $8.754$ | $8.755$ | $1.533$ | $1.534$ | $1.533$ | $1.531$ | $1.533$ | |

${\delta}_{3}$ | $-0.884$ | $-0.884$ | $-0.884$ | $-0.884$ | $-0.884$ | $0.086$ | $0.086$ | $0.086$ | $0.086$ | $0.086$ | |

${\delta}_{4}$ | $-0.108$ | $-0.108$ | $-0.108$ | $-0.108$ | $-0.108$ | $0.126$ | $0.126$ | $0.126$ | $0.126$ | $0.126$ | |

${\delta}_{5}$ | $-0.242$ | $-0.242$ | $-0.242$ | $-0.242$ | $-0.242$ | $0.122$ | $0.122$ | $0.122$ | $0.122$ | $0.122$ | |

ULOG | $\alpha $ | $5.821$ | $5.821$ | $5.821$ | $5.821$ | $5.821$ | $0.282$ | $0.282$ | $0.282$ | $0.282$ | $0.282$ |

${\delta}_{0}$ | $-3.265$ | $-3.076$ | $-2.888$ | $-2.699$ | $-2.510$ | $0.146$ | $0.145$ | $0.145$ | $0.145$ | $0.145$ | |

${\delta}_{1}$ | $0.005$ | $0.005$ | $0.005$ | $0.005$ | $0.005$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | |

${\delta}_{2}$ | $8.878$ | $8.878$ | $8.878$ | $8.878$ | $8.878$ | $0.663$ | $0.661$ | $0.662$ | $0.661$ | $0.661$ | |

${\delta}_{3}$ | $-0.932$ | $-0.932$ | $-0.932$ | $-0.932$ | $-0.932$ | $0.036$ | $0.036$ | $0.036$ | $0.036$ | $0.036$ | |

${\delta}_{4}$ | $-0.122$ | $-0.122$ | $-0.122$ | $-0.122$ | $-0.122$ | $0.051$ | $0.051$ | $0.051$ | $0.051$ | $0.051$ | |

${\delta}_{5}$ | $-0.239$ | $-0.239$ | $-0.239$ | $-0.239$ | $-0.239$ | $0.049$ | $0.049$ | $0.049$ | $0.049$ | $0.049$ | |

UWEI | $\alpha $ | $5.976$ | $5.980$ | $5.985$ | $5.992$ | $6.000$ | $0.243$ | $0.243$ | $0.243$ | $0.243$ | $0.243$ |

${\delta}_{0}$ | $-3.214$ | $-2.960$ | $-2.640$ | $-2.272$ | $-1.895$ | $0.168$ | $0.159$ | $0.148$ | $0.138$ | $0.130$ | |

${\delta}_{1}$ | $0.006$ | $0.005$ | $0.005$ | $0.005$ | $0.004$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | $0.001$ | |

${\delta}_{2}$ | $8.592$ | $8.201$ | $7.729$ | $7.218$ | $6.735$ | $0.729$ | $0.691$ | $0.647$ | $0.600$ | $0.556$ | |

${\delta}_{3}$ | $-0.966$ | $-0.919$ | $-0.863$ | $-0.803$ | $-0.746$ | $0.043$ | $0.040$ | $0.037$ | $0.035$ | $0.032$ | |

${\delta}_{4}$ | $-0.131$ | $-0.127$ | $-0.121$ | $-0.115$ | $-0.109$ | $0.055$ | $0.053$ | $0.050$ | $0.047$ | $0.044$ | |

${\delta}_{5}$ | $-0.355$ | $-0.339$ | $-0.319$ | $-0.298$ | $-0.277$ | $0.055$ | $0.053$ | $0.050$ | $0.047$ | $0.043$ |

**Table 10.**Values of the Akaike information criterion for the indicated model and quantile level ($\tau $) with arm data.

Model | $\mathit{\tau}$ Level | ||||
---|---|---|---|---|---|

0.10 | 0.25 | 0.50 | 0.75 | 0.90 | |

KUMA | $-839.660$ | $-842.224$ | $-845.030$ | $-847.657$ | $-849.813$ |

LEEG | $-846.433$ | $-849.810$ | $-853.814$ | $-858.493$ | $-863.789$ |

UBS | $-905.942$ | $-907.590$ | $-909.332$ | $-910.964$ | $-912.330$ |

UBUR | $-466.813$ | $-467.464$ | $-467.982$ | $-468.339$ | $-468.586$ |

UCHEN | $-690.483$ | $-691.734$ | $-691.722$ | $-687.523$ | $-676.318$ |

UHN | $-546.283$ | $-546.283$ | $-546.283$ | $-546.283$ | $-546.283$ |

ULOG | $-888.401$ | $-888.401$ | $-888.401$ | $-888.401$ | $-888.401$ |

UWEI | $-838.151$ | $-839.364$ | $-840.932$ | $-842.757$ | $-844.603$ |

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**MDPI and ACS Style**

Mazucheli, J.; Leiva, V.; Alves, B.; Menezes, A.F.B.
A New Quantile Regression for Modeling Bounded Data under a Unit Birnbaum–Saunders Distribution with Applications in Medicine and Politics. *Symmetry* **2021**, *13*, 682.
https://doi.org/10.3390/sym13040682

**AMA Style**

Mazucheli J, Leiva V, Alves B, Menezes AFB.
A New Quantile Regression for Modeling Bounded Data under a Unit Birnbaum–Saunders Distribution with Applications in Medicine and Politics. *Symmetry*. 2021; 13(4):682.
https://doi.org/10.3390/sym13040682

**Chicago/Turabian Style**

Mazucheli, Josmar, Víctor Leiva, Bruna Alves, and André F. B. Menezes.
2021. "A New Quantile Regression for Modeling Bounded Data under a Unit Birnbaum–Saunders Distribution with Applications in Medicine and Politics" *Symmetry* 13, no. 4: 682.
https://doi.org/10.3390/sym13040682