# A Parametric Quantile Regression Model for Asymmetric Response Variables on the Real Line

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## Abstract

**:**

## 1. Introduction

## 2. A PSN Distribution Parameterized by Its Quantile Parameter, and Its Associated Quantile Regression Model

## 3. Local Influence

#### 3.1. Case Weights Perturbation

#### 3.2. Case Response Perturbation

#### 3.3. Case Continuous Covariate Perturbation

## 4. Real Data Analysis

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Details for Score and Hessian

#### Appendix A.2. Local Influence

**Figure A1.**Index plots for ${C}_{i}({\widehat{\mathbf{\beta}}}_{1})$ (left), ${C}_{i}({\widehat{\mathbf{\beta}}}_{2})$ (center) and ${C}_{i}({\widehat{\mathbf{\beta}}}_{3})$ (right) under the weight perturbation (upper), response perturbation (center) and covariate perturbation (lower) schemes for RPSN model for $\tau =0.1$.

**Figure A2.**Index plots for ${C}_{i}\left({\widehat{\mathbf{\beta}}}_{1}\right)$ (left), ${C}_{i}\left({\widehat{\mathbf{\beta}}}_{2}\right)$ (center) and ${C}_{i}\left({\widehat{\mathbf{\beta}}}_{3}\right)$ (right) under the weight perturbation (upper), response perturbation (center) and covariate perturbation (lower) schemes for RPSN model for $\tau =0.25$.

**Figure A3.**Index plots for ${C}_{i}\left({\widehat{\mathbf{\beta}}}_{1}\right)$ (left), ${C}_{i}\left({\widehat{\mathbf{\beta}}}_{2}\right)$ (center) and ${C}_{i}\left({\widehat{\mathbf{\beta}}}_{3}\right)$ (right) under the weight perturbation (upper), response perturbation (center) and covariate perturbation (lower) schemes for RPSN model for $\tau =0.75$.

**Figure A4.**Index plots for ${C}_{i}\left({\widehat{\mathbf{\beta}}}_{1}\right)$ (left), ${C}_{i}\left({\widehat{\mathbf{\beta}}}_{2}\right)$ (center) and ${C}_{i}\left({\widehat{\mathbf{\beta}}}_{3}\right)$ (right) under the weight perturbation (upper), response perturbation (center) and covariate perturbation (lower) schemes for RPSN model for $\tau =0.9$.

**Table A1.**RCs (in %) in ML estimates and their corresponding SEs for the indicated parameter and respective p-values for the athletes dataset when observation 75 and 178 are dropped separately.

Dropped | $\mathit{\tau}$ | ||||||
---|---|---|---|---|---|---|---|

Cases | Parameter | 0.10 | 0.25 | 0.50 | 0.75 | 0.90 | |

75 | RC | 5.31 | 7.22 | 10.82 | 16.2 | 22.57 | |

RCSE | ${\beta}_{11}\left(\tau \right)$ | 0.23 | 0.20 | 0.17 | 0.11 | 0.04 | |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 1.82 | 5.03 | 10.02 | 16.09 | 22.08 | ||

RCSE | ${\beta}_{12}\left(\tau \right)$ | 0.15 | 0.05 | 0.08 | 0.07 | 0.17 | |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 6.77 | 9.84 | 14.27 | 19.20 | 23.84 | ||

RCSE | ${\beta}_{21}\left(\tau \right)$ | 0.65 | 0.93 | 1.05 | 0.71 | 0.33 | |

p-value | 0.0118 | 0.0105 | 0.0095 | 0.0086 | 0.0078 | ||

178 | RC | 0.72 | 2.62 | 6.30 | 11.88 | 18.50 | |

RC${}_{SE}$ | ${\beta}_{11}\left(\tau \right)$ | 0.17 | 0.15 | 0.12 | 0.07 | 0.00 | |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 0.12 | 3.36 | 8.60 | 14.88 | 21.06 | ||

RC${}_{SE}$ | ${\beta}_{12}\left(\tau \right)$ | 0.13 | 0.06 | 0.09 | 0.07 | 0.18 | |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 22.91 | 25.43 | 29.09 | 33.17 | 37.01 | ||

RC${}_{SE}$ | ${\beta}_{21}\left(\tau \right)$ | 0.75 | 0.47 | 0.31 | 0.61 | 1.61 | |

p-value | 0.0449 | 0.0418 | 0.0393 | 0.0371 | 0.0352 |

**Table A2.**RCs (in %) in ML estimates and their corresponding SEs for the indicated parameter and respective p-values for the athletes dataset when observations {75, 178} and {75, 162, 178} are dropped separately.

Dropped | $\mathit{\tau}$ | ||||||
---|---|---|---|---|---|---|---|

Cases | Parameter | 0.10 | 0.25 | 0.50 | 0.75 | 0.90 | |

75 and | RC | 6.30 | 8.16 | 11.69 | 17.01 | 23.29 | |

178 | RCSE | ${\beta}_{11}\left(\tau \right)$ | 0.41 | 0.39 | 0.34 | 0.28 | 0.19 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 1.75 | 5.32 | 10.58 | 16.84 | 22.97 | ||

RCSE | ${\beta}_{12}\left(\tau \right)$ | 0.29 | 0.08 | 0.03 | 0.18 | 0.27 | |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 31 | 33.34 | 36.67 | 40.38 | 43.87 | ||

RCSE | ${\beta}_{21}\left(\tau \right)$ | 0.04 | 0.27 | 0.42 | 0.11 | 0.91 | |

p-value | 0.0674 | 0.0633 | 0.0600 | 0.0572 | 0.0546 | ||

75, 162 | RC | 5.43 | 7.27 | 10.80 | 16.13 | 22.45 | |

and 178 | RCSE | ${\beta}_{11}\left(\tau \right)$ | 0.57 | 0.54 | 0.50 | 0.43 | 0.34 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 1.36 | 5.12 | 10.53 | 16.91 | 23.14 | ||

RCSE | ${\beta}_{12}\left(\tau \right)$ | 0.39 | 0.18 | 0.12 | 0.26 | 0.35 | |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 43.37 | 45.46 | 48.35 | 51.53 | 54.51 | ||

RCSE | ${\beta}_{21}\left(\tau \right)$ | 0.29 | 0.61 | 0.77 | 0.46 | 0.56 | |

p-value | 0.1300 | 0.1251 | 0.1212 | 0.1178 | 0.1149 |

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**Figure 1.**Pdf for the RPSN$(\mu =0,\sigma =1,\lambda )$ for different values of $\lambda $: $\tau =0.1$ (left panel); $\tau =0.5$ (center panel); $\tau =0.9$ (right panel). Values for $\lambda $ are: $-5$ (black line), $-1.5$ (red line), $-0.5$ (blue line), 0 (green line), $0.5$ (orange line), $1.5$ (magenta line) and 5 (purple line).

**Figure 2.**Athletes dataset: Point estimates (center line) and 95% confidence intervals (CIs) for model parameters under RPSN-QR model.

**Figure 3.**Data analysis: Fitted RPSN-QR model lines for the response (left panel for males, center panel for females) and scale parameter (right panel) over the grid $\tau =\{0.10,0.25,0.50,0.75,0.90\}$.

**Figure 4.**Index plots for ${C}_{i}({\widehat{\mathbf{\beta}}}_{1})$ (left), ${C}_{i}({\widehat{\mathbf{\beta}}}_{2})$ (center) and ${C}_{i}({\widehat{\mathbf{\beta}}}_{3})$ (right) under the weight perturbation (upper), response perturbation (center) and covariate perturbation (lower) schemes for RPSN model for $\tau =0.5$.

Coefficient | SN | PN | PSN |
---|---|---|---|

Skewness | (−0.9953, 0.9953) | [−0.6115, 0.9007] | [−1.6476, 0.9953) |

Kurtosis | [3, 3.8692) | [1.7170, 4.3556] | [1.4672, 5.4386] |

$\mathit{\tau}$ | SKN | SKT | GSC | RPSN ($\mathit{\sigma}$ Constant) | RPSN (Modeling $\mathit{\sigma}$) |
---|---|---|---|---|---|

0.10 | 1097.74 | 817.77 | 803.08 | 808.64 | 801.37 |

0.25 | 1084.46 | 803.90 | 801.96 | 811.08 | 803.11 |

0.50 | 1095.56 | 810.99 | 854.38 | 815.79 | 806.76 |

0.75 | 1151.40 | 854.57 | 861.37 | 824.56 | 814.01 |

0.90 | 1220.96 | 914.43 | 865.16 | 838.78 | 825.95 |

**Table 3.**Estimates and SE for parameters in athletes dataset in RPSN-quantile regression (QR) model for different values of $\tau $.

$\mathit{\tau}=0.10$ | $\mathit{\tau}=0.50$ | $\mathit{\tau}=0.90$ | |||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | Est. | SE | p-Value | Est. | SE | p-Value | Est. | SE | p-Value |

${\beta}_{10}\left(\tau \right)$ | 6.4642 | 1.1552 | - | 6.7727 | 1.0867 | - | 6.1798 | 1.0859 | - |

${\beta}_{11}\left(\tau \right)$ | 2.3077 | 0.3742 | <0.0001 | 2.5008 | 0.3728 | <0.0001 | 2.9324 | 0.3695 | <0.0001 |

${\beta}_{12}\left(\tau \right)$ | 0.2037 | 0.0157 | <0.0001 | 0.2299 | 0.0147 | <0.0001 | 0.2728 | 0.0151 | <0.0001 |

${\beta}_{20}\left(\tau \right)$ | 0.7633 | 0.7952 | - | 0.2252 | 0.3996 | - | −0.6261 | 0.2700 | - |

${\beta}_{21}\left(\tau \right)$ | 0.0096 | 0.0036 | 0.0040 | 0.0108 | 0.0037 | 0.0017 | 0.0125 | 0.0035 | 0.0002 |

${\beta}_{30}\left(\tau \right)$ | −1.1940 | 1.4984 | - | −0.8381 | 0.5984 | - | −0.4916 | 0.2588 | - |

**Table 4.**p-values for normality K-S test for residuals under our RPSN-QR model for the athletes dataset for different quantiles $\tau $’s.

$\mathit{\tau}$ | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50 |

p-value | 0.995 | 0.996 | 0.991 | 0.976 | 0.951 | 0.914 | 0.864 | 0.853 | 0.837 |

$\tau $ | 0.55 | 0.60 | 0.65 | 0.70 | 0.75 | 0.80 | 0.85 | 0.90 | |

p-value | 0.765 | 0.810 | 0.777 | 0.683 | 0.604 | 0.524 | 0.394 | 0.191 |

**Table 5.**Relative changes (RC) (in %) in ML estimates and their corresponding SE’s for the indicated parameter and respective p-values for the athletes dataset when observations 53, 75, 162 and 178 are dropped.

Dropped | $\mathit{\tau}$ | ||||||
---|---|---|---|---|---|---|---|

Cases | Parameter | 0.10 | 0.25 | 0.50 | 0.75 | 0.90 | |

53, 75, | RC | 7.20 | 9.06 | 12.56 | 17.82 | 24.05 | |

162 and 178 | RCSE | ${\beta}_{11}\left(\tau \right)$ | 0.74 | 0.74 | 0.69 | 0.63 | 0.52 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 1.98 | 5.63 | 10.93 | 17.22 | 23.38 | ||

RCSE | ${\beta}_{12}\left(\tau \right)$ | 0.47 | 0.26 | 0.20 | 0.34 | 0.42 | |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | ||

RC | 34.55 | 36.83 | 40.05 | 43.61 | 46.96 | ||

RCSE | ${\beta}_{21}\left(\tau \right)$ | 0.23 | 0.53 | 0.69 | 0.36 | 0.66 | |

p-value | 0.0806 | 0.0762 | 0.0727 | 0.0697 | 0.0670 |

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**MDPI and ACS Style**

Gallardo, D.I.; Bourguignon, M.; Galarza, C.E.; Gómez, H.W.
A Parametric Quantile Regression Model for Asymmetric Response Variables on the Real Line. *Symmetry* **2020**, *12*, 1938.
https://doi.org/10.3390/sym12121938

**AMA Style**

Gallardo DI, Bourguignon M, Galarza CE, Gómez HW.
A Parametric Quantile Regression Model for Asymmetric Response Variables on the Real Line. *Symmetry*. 2020; 12(12):1938.
https://doi.org/10.3390/sym12121938

**Chicago/Turabian Style**

Gallardo, Diego I., Marcelo Bourguignon, Christian E. Galarza, and Héctor W. Gómez.
2020. "A Parametric Quantile Regression Model for Asymmetric Response Variables on the Real Line" *Symmetry* 12, no. 12: 1938.
https://doi.org/10.3390/sym12121938