# An Asymmetric Bimodal Distribution with Application to Quantile Regression

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## Abstract

**:**

## 1. Introduction

## 2. Gamma–Sinh Cauchy Distribution

- $\varphi =1$ SC distribution,

**Proposition**

**1.**

**Proof.**

#### The GSC Model for Quantile Regression

`uniroot`function can be used. Table 1 shows some values for $\varphi \left(q\right)$ with different values for q.

## 3. ML Estimation for the GSC Distribution

#### 3.1. ML Estimation

`NumDeriv`routine with the R software (R Core Team [22]).

#### 3.2. Simulation Study

## 4. Applications

#### 4.1. Application 1: Without Covariates

#### 4.2. Data Set 2: Quantile Regression to Bimodal Data

`sn`package in R. This data set is related to 102 male and 100 female athletes collected at the Australian Institute of Sport. The linear model considered is

## 5. Final Comments

- The GSC distribution contains the SC and hyperbolic secant models as special cases.
- The GSC distribution presents great flexibility in its modes, as can be observed in Figure 1.
- The proposed model has a closed-form expression for its cdf.
- In the two applications, we show that the GSC model fits better than the other models.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Plots for the gamma–sinh Cauchy (GSC) model for different values of the parameters with $\mu =0$, $\sigma =1$ (

**a**) $\varphi =1$ (

**b**) $\lambda =0.5$, and (

**c**,

**d**) $\lambda =0.2$.

**Figure 3.**Skewness and kurtosis coefficients for the GSC $(\varphi ,\lambda ,\mu ,\sigma )$ model with different values for $\lambda $ and $\varphi $.

**Figure 5.**Estimates for regression coefficients (and 95% confidence interval)s for variables

`bmi`(left panel) and

`lbm`(right panel) in different quantile regression models with quantiles equal to $0.1,0.25,0.5,0.75$, and $0.9$ and response variable

`Bfat`.

**Figure 6.**Distribution for $0.1$ and $0.75$ quantiles of body fat percentage considering body mass index and lean body mass equal to 22.96 and 64.87, respectively. Curves in black, red, and green represent the density functions estimated by the GSC, SKL, and SKT models, respectively.

q | $0.1$ | $0.2$ | $0.3$ | $0.4$ | $0.5$ | $0.6$ | $0.7$ | $0.8$ | $0.9$ |

$\mathit{\varphi}\left(\mathit{q}\right)$ | 2.301 | 1.802 | 1.475 | 1.219 | 1.000 | 0.802 | 0.613 | 0.427 | 0.230 |

$\mathit{n}=100$ | $\mathit{n}=200$ | $\mathit{n}=500$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\varphi}$ | $\mathit{\lambda}$ | parameter | bias | RMSE | CP | bias | RMSE | CP | bias | RMSE | CP |

0.75 | 0.5 | $\mu $ | −0.094 | 0.663 | 0.846 | −0.055 | 0.578 | 0.900 | −0.014 | 0.457 | 0.932 |

$\sigma $ | −0.037 | 0.392 | 0.880 | −0.014 | 0.337 | 0.904 | −0.006 | 0.268 | 0.933 | ||

$\lambda $ | −0.031 | 0.372 | 0.869 | −0.013 | 0.317 | 0.905 | −0.006 | 0.252 | 0.930 | ||

$\varphi $ | 0.044 | 0.411 | 0.867 | 0.023 | 0.349 | 0.907 | 0.006 | 0.272 | 0.934 | ||

1.0 | $\mu $ | −0.015 | 0.607 | 0.831 | 0.021 | 0.532 | 0.880 | 0.016 | 0.428 | 0.925 | |

$\sigma $ | −0.035 | 0.460 | 0.848 | −0.024 | 0.396 | 0.886 | −0.009 | 0.320 | 0.927 | ||

$\lambda $ | −0.004 | 0.544 | 0.831 | −0.010 | 0.459 | 0.889 | −0.003 | 0.365 | 0.927 | ||

$\varphi $ | 0.018 | 0.424 | 0.843 | −0.004 | 0.363 | 0.890 | −0.006 | 0.290 | 0.926 | ||

2.0 | $\mu $ | 0.011 | 0.361 | 0.934 | 0.004 | 0.298 | 0.944 | 0.001 | 0.233 | 0.944 | |

$\sigma $ | −0.011 | 0.504 | 0.911 | −0.007 | 0.419 | 0.932 | −0.003 | 0.333 | 0.945 | ||

$\lambda $ | 0.055 | 0.789 | 0.928 | 0.017 | 0.650 | 0.939 | 0.010 | 0.513 | 0.945 | ||

$\varphi $ | −0.007 | 0.337 | 0.940 | −0.003 | 0.279 | 0.949 | −0.001 | 0.220 | 0.947 | ||

1.0 | 0.5 | $\mu $ | −0.060 | 0.666 | 0.846 | −0.036 | 0.580 | 0.899 | −0.012 | 0.460 | 0.933 |

$\sigma $ | −0.044 | 0.373 | 0.877 | −0.022 | 0.318 | 0.915 | −0.009 | 0.253 | 0.933 | ||

$\lambda $ | −0.033 | 0.366 | 0.867 | −0.018 | 0.308 | 0.913 | −0.007 | 0.245 | 0.936 | ||

$\varphi $ | 0.045 | 0.458 | 0.874 | 0.023 | 0.388 | 0.909 | 0.007 | 0.304 | 0.938 | ||

1.0 | $\mu $ | −0.052 | 0.622 | 0.816 | −0.024 | 0.543 | 0.881 | −0.002 | 0.428 | 0.940 | |

$\sigma $ | −0.040 | 0.431 | 0.832 | −0.024 | 0.367 | 0.887 | −0.006 | 0.293 | 0.936 | ||

$\lambda $ | −0.017 | 0.535 | 0.811 | −0.015 | 0.448 | 0.875 | 0.000 | 0.354 | 0.934 | ||

$\varphi $ | 0.060 | 0.493 | 0.842 | 0.027 | 0.417 | 0.890 | 0.005 | 0.324 | 0.946 | ||

2.0 | $\mu $ | −0.001 | 0.357 | 0.937 | 0.000 | 0.291 | 0.946 | −0.001 | 0.229 | 0.951 | |

$\sigma $ | −0.014 | 0.494 | 0.896 | −0.006 | 0.413 | 0.929 | 0.000 | 0.328 | 0.942 | ||

$\lambda $ | 0.033 | 0.795 | 0.916 | 0.017 | 0.658 | 0.937 | 0.011 | 0.521 | 0.944 | ||

$\varphi $ | 0.007 | 0.372 | 0.942 | 0.002 | 0.303 | 0.947 | 0.002 | 0.238 | 0.949 | ||

1.5 | 0.5 | $\mu $ | 0.015 | 0.683 | 0.850 | 0.014 | 0.597 | 0.894 | 0.008 | 0.480 | 0.925 |

$\sigma $ | −0.045 | 0.354 | 0.890 | −0.021 | 0.300 | 0.926 | −0.009 | 0.239 | 0.935 | ||

$\lambda $ | −0.022 | 0.366 | 0.877 | −0.014 | 0.308 | 0.916 | −0.006 | 0.245 | 0.937 | ||

$\varphi $ | 0.026 | 0.533 | 0.876 | 0.008 | 0.455 | 0.906 | 0.002 | 0.364 | 0.930 | ||

1.0 | $\mu $ | −0.134 | 0.688 | 0.835 | −0.138 | 0.612 | 0.856 | −0.076 | 0.492 | 0.904 | |

$\sigma $ | -0.038 | 0.413 | 0.836 | −0.025 | 0.347 | 0.856 | −0.011 | 0.275 | 0.896 | ||

$\lambda $ | 0.091 | 0.581 | 0.840 | −0.001 | 0.459 | 0.860 | −0.010 | 0.360 | 0.896 | ||

$\varphi $ | 0.211 | 0.681 | 0.886 | 0.166 | 0.579 | 0.880 | 0.076 | 0.446 | 0.912 | ||

2.0 | $\mu $ | −0.029 | 0.383 | 0.933 | −0.009 | 0.309 | 0.945 | −0.004 | 0.241 | 0.948 | |

$\sigma $ | −0.014 | 0.472 | 0.895 | −0.007 | 0.395 | 0.927 | −0.003 | 0.313 | 0.942 | ||

$\lambda $ | 0.065 | 0.784 | 0.932 | 0.023 | 0.646 | 0.942 | 0.010 | 0.510 | 0.952 | ||

$\varphi $ | 0.070 | 0.484 | 0.948 | 0.023 | 0.379 | 0.954 | 0.009 | 0.294 | 0.950 |

n | $\overline{\mathit{y}}$ | ${\mathit{s}}^{2}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ |
---|---|---|---|---|

1150 | 3.535 | 0.422 | −0.986 | 4.855 |

**Table 4.**Maximum likelihood (ML) estimates for the GSC, exponentiated sinh Cauchy (ECG), and skew-normal (SN) models for the roller data set.

Parameter | GSC | ECG | SN |
---|---|---|---|

$\mu $ | 4.1115 (0.0388) | 4.0460 (0.0482) | 4.2475 (0.0276) |

$\sigma $ | 0.2053 (0.0172) | 0.1903 (0.0205) | 0.9644 (0.0286) |

$\lambda $ | 0.5353 (0.0682) | 0.5535 (0.0860) | −2.7578 (0.2426) |

$\varphi $ | 0.3621 (0.0373) | 0.3322 (0.0445) | − |

log-likelihood | −1053.8 | −1056.0 | −1071.3 |

AIC | 2115.7 | 2119.9 | 2148.7 |

**Table 5.**AIC and p-value for K–S test in the ais data set for the GSC, skewed Laplace (SKL), and skewed Student-t (SKT) models and different quantiles.

AIC | p-value for K–S test | |||||
---|---|---|---|---|---|---|

q | GSC | SKL | SKT | GSC | SKL | SKT |

$0.10$ | 1156.54 | 1194.28 | 1166.74 | 0.79 | 0.06 | 0.02 |

$0.25$ | 1160.72 | 1172.70 | 1153.15 | 0.65 | 0.71 | 0.96 |

$0.50$ | 1162.74 | 1182.66 | 1161.80 | 0.32 | 0.13 | 0.31 |

$0.75$ | 1159.50 | 1221.65 | 1200.87 | 0.87 | <0.001 | 0.02 |

$0.90$ | 1211.71 | 1280.45 | 1253.48 | 0.02 | <0.001 | <0.001 |

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**MDPI and ACS Style**

Gómez, Y.M.; Gómez-Déniz, E.; Venegas, O.; Gallardo, D.I.; Gómez, H.W.
An Asymmetric Bimodal Distribution with Application to Quantile Regression. *Symmetry* **2019**, *11*, 899.
https://doi.org/10.3390/sym11070899

**AMA Style**

Gómez YM, Gómez-Déniz E, Venegas O, Gallardo DI, Gómez HW.
An Asymmetric Bimodal Distribution with Application to Quantile Regression. *Symmetry*. 2019; 11(7):899.
https://doi.org/10.3390/sym11070899

**Chicago/Turabian Style**

Gómez, Yolanda M., Emilio Gómez-Déniz, Osvaldo Venegas, Diego I. Gallardo, and Héctor W. Gómez.
2019. "An Asymmetric Bimodal Distribution with Application to Quantile Regression" *Symmetry* 11, no. 7: 899.
https://doi.org/10.3390/sym11070899