# An Asymmetric Bimodal Double Regression Model

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## Abstract

**:**

## 1. Introduction

## 2. The GSC Regression Model

**Proposition**

**1.**

**Proof.**

- For $0<\lambda <1/\sqrt{2}$, the equation ${f}^{\prime}\left(x\right)=0$ has three solutions. In this case, ${f}^{\prime}\left(x\right)>0$, $\forall x\in (-\infty ,\mathrm{asinh}\left(\right)open="("\; close=")">-\frac{\sqrt{1-2{\lambda}^{2}}}{\lambda})$ and ${f}^{\prime}\left(x\right)<0$, $\forall x\in (\mathrm{asinh}\left(\right)open="("\; close=")">-\frac{\sqrt{1-2{\lambda}^{2}}}{\lambda},0)$. Then, $\mathrm{asinh}\left(\right)open="("\; close=")">\pm \frac{\sqrt{1-2{\lambda}^{2}}}{\lambda}$ are the two modes of the distribution.
- For $\lambda \ge 1/\sqrt{2}$, the equation ${f}^{\prime}\left(x\right)=0$ has one solution. In this case, ${f}^{\prime}\left(x\right)>0$, $\forall x<0$ and ${f}^{\prime}\left(x\right)<0$, $\forall x>0$. Then, $x=0$ is the only mode of the distribution.

## 3. Simulation Study

## 4. Application

`sn`in R [28], which includes data on 202 athletes collected at the Australian Institute of Sport. Codes were performed in [27] and are available upon request. Our main aim is to explain the body fat percentage (

`Bfat`) in terms of the body mass index (

`bmi`) and the lean body mass (

`lbm`). Particularly, we consider

`Bfat`${}_{i}\left(q\right)\sim GSC({\mu}_{i}\left(q\right),{\sigma}_{i}\left(q\right),\lambda ,\varphi \left(q\right))$, where $\varphi \left(q\right)$ satisfies (1), $q\in (0,1)$ and for $i=1,\dots ,202$, we have that

`bmi`and

`lbm`explain both the q-th quantile of

`Bfat`and the scale of the distribution. The same structure of covariates was considered in [13], but without modeling the scale parameter; i.e., considering ${\beta}_{22}\left(q\right)={\beta}_{23}\left(q\right)=0$. We refer to those models as GSC and GSC${}_{0}$ for the cases where $\sigma $ is modeled and not modeled, respectively. We considered $q\in \{0.1,0.25,0.5,0.75,0.9\}$. Our approach is compared with the skewed Laplace (SKL) model in [29]. Table 2 shows the Akaike information criterion (AIC [30]) for the three models. We also present the statistic for the likelihood ratio test (LRT) to test ${H}_{0}:{\beta}_{22}\left(q\right)={\beta}_{23}\left(q\right)=0$ versus ${H}_{0}:{\beta}_{22}\left(q\right)\ne 0$ or ${\beta}_{23}\left(q\right)\ne 0$, for all the quantiles considered. In addition, we also compute the quantile residuals [31] for the GSC model. If the model is correctly specified, the residual should be a random sample from the standard normal distribution. We checked this assumption with the traditional Kolmogorov–Smirnov test. Note that the GSC presents the lowest AIC for the quantiles up to the median, and GSC${}_{0}$ presents the lowest AIC for the rest of the quantiles. This is explained because, according to the LRT, the coefficients related to the

`bmi`and

`lbm`variables are not significant (under any common level of significance) for modeling the scale parameter for $q=0.75$ and $q=0.9$, while they are significant for the rest of the quantiles. Finally, based on the quantile residuals, the GSC double regression model seems to be appropriate for modeling all quantiles, except the largest. Figure 2 also shows the regression coefficients in terms of the quantiles and their respective 95% confidence intervals. Note that ${\beta}_{12}\left(q\right)$ and ${\beta}_{13}\left(q\right)$ are significant (based on 5% significance) for all the quantiles considered; i.e., the

`bmi`and

`lbm`variables are relevant for explaining the different quantiles of

`Bfat`. Specifically, we can obtain the following interpretations for ${\beta}_{12}\left(q\right)$ and ${\beta}_{13}\left(q\right)$:

- (Interpreting ${\beta}_{12}\left(q\right)$) For a fixed
`lbm`, for athletes in the lowest 10% of`Bfat`, the`Bfat`is increased by 0.8572 units (95% confidence interval 0.4191; 1.2954) for each unit increase in`bmi`, and for athletes in the highest 90% of`Bfat`, the`Bfat`is increased by 2.5834 units (95% confidence interval 2.3227; 2.8442) for each unit increase in`bmi`. - (Interpreting ${\beta}_{13}\left(q\right)$) For a fixed
`bmi`, for athletes in the lowest 10% of`Bfat`, the`Bfat`is decreased by 0.3039 units (95% confidence interval −0.4093; −0.1984) for each unit increase in`lbm`, and for athletes in the highest 10% of`Bfat`, the`Bfat`is decreased by 0.5511 units (95% confidence interval −0.6077; −0.4945) for each unit increase in`lbm`.

`Bfat`.

`Bfat`under different scenarios for

`bmi`and

`lbm`. We note that these estimated pdf values assumed different shapes—unimodal, bimodal symmetric, and bimodal asymmetric—justifying the use of the double regression GSC model in this example. Finally, Figure 4 shows the pairs $(\lambda ,q)$ for the five quantiles modeled, identifying the unimodal and bimodal cases.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Relation between q and $\varphi $ and (

**b**) regions of unimodality and bimodality for the GSC model in terms of q and $\lambda $.

**Figure 2.**Estimated parameters for regression coefficients (and 95% confidence intervals) for different quantile regression models in the athlete data set.

**Figure 3.**Estimated density function for different quantiles of

`Bfat`under different combinations of

`bmi`and

`lbm`: (

**a**) $q=0.25,\mathtt{bmi}=32$,

`lbm`= 40; (

**b**) $q=0.50,\mathtt{bmi}=32$,

`lbm`= 40; (

**c**) $q=0.25,\mathtt{bmi}=30$,

`lbm`= 80 and; (

**d**) $q=0.75,\mathtt{bmi}=30$,

`lbm`= 80.

**Figure 4.**Points of unimodality and bimodality for the GSC model in the athlete data set for the different quantiles modeled.

**Table 1.**Estimated bias, SE, RMSE, and 95%CP for the ML estimators for the GSC double regression model under different scenarios based on 5000 Monte Carlo replicates.

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

q | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | $log\left(\mathit{\lambda}\right)$ | Parameter | Bias | SE | RMSE | CP | Bias | SE | RMSE | CP | Bias | SE | RMSE | CP |

0.10 | (1, −1, 0.5) | (−1, 1.6, −0.5) | −1.39 | ${\beta}_{10}$ | 0.0146 | 0.0514 | 0.0648 | 0.8694 | 0.0066 | 0.0329 | 0.0363 | 0.9198 | 0.0025 | 0.0200 | 0.0208 | 0.9394 |

${\beta}_{11}$ | 0.0060 | 0.0275 | 0.0340 | 0.8786 | 0.0029 | 0.0177 | 0.0191 | 0.9246 | 0.0010 | 0.0106 | 0.0110 | 0.9398 | ||||

${\beta}_{12}$ | −0.0011 | 0.0078 | 0.0099 | 0.8676 | −0.0003 | 0.0058 | 0.0066 | 0.9086 | −0.0002 | 0.0038 | 0.0040 | 0.9410 | ||||

${\beta}_{20}$ | −0.0544 | 0.1004 | 0.1184 | 0.9006 | −0.0257 | 0.0699 | 0.0750 | 0.9314 | −0.0085 | 0.0438 | 0.0449 | 0.9424 | ||||

${\beta}_{21}$ | 0.0194 | 0.0505 | 0.0598 | 0.9040 | 0.0083 | 0.0357 | 0.0385 | 0.9314 | 0.0033 | 0.0224 | 0.0236 | 0.9390 | ||||

${\beta}_{22}$ | −0.0089 | 0.0457 | 0.0503 | 0.9222 | −0.0051 | 0.0332 | 0.0359 | 0.9268 | −0.0009 | 0.0206 | 0.0210 | 0.9462 | ||||

$log\left(\lambda \right)$ | −0.0983 | 0.3151 | 0.3520 | 0.9332 | −0.0498 | 0.2163 | 0.2273 | 0.9460 | −0.0162 | 0.1340 | 0.1360 | 0.9498 | ||||

1.61 | ${\beta}_{10}$ | 0.0003 | 0.0449 | 0.0536 | 0.8920 | −0.0002 | 0.0287 | 0.0310 | 0.9310 | 0.0002 | 0.0175 | 0.0179 | 0.9428 | |||

${\beta}_{11}$ | 0.0003 | 0.0255 | 0.0311 | 0.8866 | −0.0001 | 0.0164 | 0.0181 | 0.9222 | 0.0001 | 0.0099 | 0.0102 | 0.9410 | ||||

${\beta}_{12}$ | −0.0001 | 0.0069 | 0.0098 | 0.8308 | 0.0001 | 0.0053 | 0.0064 | 0.8950 | 0.0000 | 0.0036 | 0.0039 | 0.9334 | ||||

${\beta}_{20}$ | −0.0627 | 0.1116 | 0.1311 | 0.8904 | −0.0315 | 0.0791 | 0.0863 | 0.9162 | −0.0113 | 0.0501 | 0.0514 | 0.9398 | ||||

${\beta}_{21}$ | 0.0113 | 0.0460 | 0.0504 | 0.9264 | 0.0047 | 0.0337 | 0.0352 | 0.9386 | 0.0014 | 0.0215 | 0.0216 | 0.9472 | ||||

${\beta}_{22}$ | −0.0058 | 0.0467 | 0.0503 | 0.9280 | −0.0023 | 0.0336 | 0.0350 | 0.9388 | −0.0012 | 0.0211 | 0.0215 | 0.9464 | ||||

$log\left(\lambda \right)$ | −0.1492 | 0.3012 | 0.3461 | 0.9266 | −0.0777 | 0.2073 | 0.2277 | 0.9312 | −0.0274 | 0.1288 | 0.1333 | 0.9446 | ||||

0.50 | (−0.5, −2, 1) | (−1, 1.6, −0.5) | −1.39 | ${\beta}_{10}$ | 0.0126 | 0.0512 | 0.0636 | 0.8744 | 0.0073 | 0.0331 | 0.0368 | 0.9182 | 0.0023 | 0.0200 | 0.0204 | 0.9398 |

${\beta}_{11}$ | 0.0050 | 0.0275 | 0.0338 | 0.8812 | 0.0032 | 0.0178 | 0.0195 | 0.9230 | 0.0009 | 0.0106 | 0.0108 | 0.9422 | ||||

${\beta}_{12}$ | −0.0011 | 0.0077 | 0.0098 | 0.8628 | −0.0005 | 0.0058 | 0.0067 | 0.9118 | −0.0003 | 0.0039 | 0.0041 | 0.9350 | ||||

${\beta}_{20}$ | −0.0576 | 0.1001 | 0.1197 | 0.8924 | −0.0239 | 0.0700 | 0.0748 | 0.9288 | −0.0087 | 0.0438 | 0.0446 | 0.9446 | ||||

${\beta}_{21}$ | 0.0186 | 0.0499 | 0.0586 | 0.9074 | 0.0080 | 0.0358 | 0.0381 | 0.9342 | 0.0036 | 0.0224 | 0.0236 | 0.9376 | ||||

${\beta}_{22}$ | −0.0084 | 0.0455 | 0.0489 | 0.9274 | −0.0044 | 0.0333 | 0.0352 | 0.9346 | −0.0011 | 0.0207 | 0.0207 | 0.9492 | ||||

$log\left(\lambda \right)$ | −0.1142 | 0.3149 | 0.3507 | 0.9382 | −0.0432 | 0.2162 | 0.2242 | 0.9448 | −0.0154 | 0.1340 | 0.1391 | 0.9422 | ||||

(1, 0.7, −0.3) | −1.39 | ${\beta}_{10}$ | −0.0068 | 0.0529 | 0.0668 | 0.8808 | −0.0020 | 0.0346 | 0.0384 | 0.9198 | −0.0008 | 0.0210 | 0.0220 | 0.9428 | ||

${\beta}_{11}$ | −0.0025 | 0.0297 | 0.0382 | 0.8720 | −0.0005 | 0.0196 | 0.0223 | 0.9164 | −0.0003 | 0.0118 | 0.0124 | 0.9428 | ||||

${\beta}_{12}$ | 0.0002 | 0.0082 | 0.0121 | 0.8276 | 0.0001 | 0.0063 | 0.0079 | 0.8782 | 0.0000 | 0.0043 | 0.0047 | 0.9342 | ||||

${\beta}_{20}$ | −0.0667 | 0.1081 | 0.1297 | 0.8862 | −0.0275 | 0.0765 | 0.0813 | 0.9306 | −0.0102 | 0.0482 | 0.0497 | 0.9402 | ||||

${\beta}_{21}$ | 0.0154 | 0.0511 | 0.0574 | 0.9182 | 0.0048 | 0.0375 | 0.0393 | 0.9352 | 0.0023 | 0.0238 | 0.0245 | 0.9430 | ||||

${\beta}_{22}$ | −0.0070 | 0.0512 | 0.0560 | 0.9208 | −0.0047 | 0.0371 | 0.0386 | 0.9426 | −0.0010 | 0.0232 | 0.0234 | 0.9510 | ||||

$log\left(\lambda \right)$ | −0.1579 | 0.3161 | 0.3625 | 0.9240 | −0.0694 | 0.2164 | 0.2292 | 0.9398 | −0.0227 | 0.1342 | 0.1375 | 0.9440 | ||||

0.75 | (1, −1, 0.5) | (1, 0.7, −0.3) | 1.61 | ${\beta}_{10}$ | −0.0061 | 0.5813 | 0.6422 | 0.9130 | −0.0004 | 0.4112 | 0.4314 | 0.9352 | −0.0020 | 0.2552 | 0.2604 | 0.9442 |

${\beta}_{11}$ | 0.0032 | 0.3709 | 0.4267 | 0.9004 | 0.0015 | 0.2696 | 0.2875 | 0.9296 | 0.0000 | 0.1632 | 0.1668 | 0.9426 | ||||

${\beta}_{12}$ | 0.0059 | 0.2288 | 0.2930 | 0.8690 | 0.0061 | 0.1650 | 0.1863 | 0.9152 | −0.0001 | 0.1074 | 0.1115 | 0.9414 | ||||

${\beta}_{20}$ | −0.0629 | 0.1118 | 0.1297 | 0.8912 | −0.0292 | 0.0792 | 0.0866 | 0.9180 | −0.0127 | 0.0500 | 0.0510 | 0.9422 | ||||

${\beta}_{21}$ | 0.0079 | 0.0458 | 0.0497 | 0.9274 | 0.0033 | 0.0338 | 0.0351 | 0.9400 | 0.0019 | 0.0214 | 0.0219 | 0.9490 | ||||

${\beta}_{22}$ | −0.0037 | 0.0467 | 0.0503 | 0.9292 | −0.0012 | 0.0337 | 0.0354 | 0.9382 | −0.0007 | 0.0211 | 0.0213 | 0.9462 | ||||

$log\left(\lambda \right)$ | −0.1481 | 0.3016 | 0.3408 | 0.9286 | −0.0727 | 0.2073 | 0.2253 | 0.9362 | −0.0313 | 0.1289 | 0.1305 | 0.9508 | ||||

(−0.5, −2, 1) | (1, 0.7, −0.3) | 1.61 | ${\beta}_{10}$ | −0.0043 | 0.1204 | 0.1302 | 0.9226 | −0.0013 | 0.0834 | 0.0886 | 0.9322 | −0.0007 | 0.0516 | 0.0519 | 0.9474 | |

${\beta}_{11}$ | 0.0026 | 0.0772 | 0.0880 | 0.9102 | 0.0009 | 0.0547 | 0.0594 | 0.9242 | 0.0005 | 0.0330 | 0.0333 | 0.9462 | ||||

${\beta}_{12}$ | −0.0013 | 0.0494 | 0.0627 | 0.8624 | −0.0006 | 0.0338 | 0.0366 | 0.9208 | 0.0001 | 0.0217 | 0.0224 | 0.9370 | ||||

${\beta}_{20}$ | −0.0547 | 0.2266 | 0.2380 | 0.9246 | −0.0307 | 0.1588 | 0.1659 | 0.9306 | −0.0099 | 0.1000 | 0.0997 | 0.9500 | ||||

${\beta}_{21}$ | 0.0264 | 0.1123 | 0.1207 | 0.9282 | 0.0098 | 0.0805 | 0.0833 | 0.9414 | 0.0045 | 0.0504 | 0.0517 | 0.9428 | ||||

${\beta}_{22}$ | −0.0142 | 0.1147 | 0.1221 | 0.9342 | −0.0080 | 0.0803 | 0.0843 | 0.9382 | −0.0014 | 0.0496 | 0.0508 | 0.9470 | ||||

$log\left(\lambda \right)$ | −0.0064 | 0.2947 | 0.2999 | 0.9452 | −0.0107 | 0.2041 | 0.2084 | 0.9432 | −0.0028 | 0.1282 | 0.1277 | 0.9488 |

**Table 2.**AIC for GSC, GSC${}_{0}$, and SKL models in the athlete data set for different quantiles. We also present the statistical p-value.

AIC | log-Likelihood | LRT | KS | |||||
---|---|---|---|---|---|---|---|---|

$\mathit{\tau}$ | GSC${}_{0}$ | GSC | SKL | GSC${}_{0}$ | GSC | Statistical | p-Value | p-Value |

0.10 | 1168.54 | 1154.64 | 1194.28 | −574.27 | −563.32 | 21.90 | <0.0001 | 0.988 |

0.25 | 1172.72 | 1164.55 | 1172.70 | −576.36 | −568.27 | 16.17 | 0.0003 | 0.646 |

0.50 | 1174.74 | 1171.83 | 1182.66 | −577.37 | −571.91 | 10.91 | 0.0043 | 0.180 |

0.75 | 1171.50 | 1176.51 | 1221.65 | −575.75 | −574.25 | 3.00 | 0.2235 | 0.839 |

0.90 | 1223.71 | 1229.75 | 1280.45 | −601.85 | −600.88 | 1.95 | 0.3768 | 0.004 |

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**MDPI and ACS Style**

Gómez, Y.M.; Gallardo, D.I.; Venegas, O.; Magalhães, T.M.
An Asymmetric Bimodal Double Regression Model. *Symmetry* **2021**, *13*, 2279.
https://doi.org/10.3390/sym13122279

**AMA Style**

Gómez YM, Gallardo DI, Venegas O, Magalhães TM.
An Asymmetric Bimodal Double Regression Model. *Symmetry*. 2021; 13(12):2279.
https://doi.org/10.3390/sym13122279

**Chicago/Turabian Style**

Gómez, Yolanda M., Diego I. Gallardo, Osvaldo Venegas, and Tiago M. Magalhães.
2021. "An Asymmetric Bimodal Double Regression Model" *Symmetry* 13, no. 12: 2279.
https://doi.org/10.3390/sym13122279