# The Arctan Power Distribution: Properties, Quantile and Modal Regressions with Applications to Biomedical Data

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## Abstract

**:**

## 1. Introduction

- Develop a flexible unit distribution that is able to model data that are left-skewed, right-skewed, symmetric, J, and reversed-J shapes.
- Develop a unit distribution capable of modeling data with increasing, bathtub, and modified upside-down bathtub hazard rate functions (HRFs).
- Develop quantile regression for modeling response variables that are skewed or contain extreme values.
- Develop modal regression for modeling response variables that are asymmetric or heavy-tailed.

## 2. Development of AP Distribution

## 3. Some Statistical Properties

#### 3.1. Mode

**Proposition**

**1.**

**Proof.**

#### 3.2. Quantile Function

**Proposition**

**2.**

**Proof.**

#### 3.3. Moments and Generating Function

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

#### 3.4. Order Statistics

## 4. Bivariate AP Distribution

- (a)
- $\alpha =8.5,\beta =2.5,{\rho}_{1}=0.4,{\rho}_{2}=0.1,{\rho}_{3}=0.2$,
- (b)
- $\alpha =4.5,\beta =8.2,{\rho}_{1}=-0.3,{\rho}_{2}=0.4,{\rho}_{3}=-0.2$ and
- (c)
- $\alpha =3.4,\beta =6.2,{\rho}_{1}=0.3,{\rho}_{2}=0.4,{\rho}_{3}=-0.6$.

- (a)
- $\alpha =8.5,\beta =2.5,{\rho}_{1}=0.4,{\rho}_{2}=0.1,{\rho}_{3}=0.2$,
- (b)
- $\alpha =4.5,\beta =8.2,{\rho}_{1}=-0.3,{\rho}_{2}=0.4,{\rho}_{3}=-0.2$ and
- (c)
- $\alpha =3.4,\beta =2.5,{\rho}_{1}=0.3,{\rho}_{2}=0.4,{\rho}_{3}=-0.6$.

## 5. Estimation Methods and Simulations

#### 5.1. Maximum Likelihood Estimation

#### 5.2. Ordinary and Weighted Least Squares Estimation

#### 5.3. Cramér–Von Mises Estimation

#### 5.4. Anderson–Darling Estimation

#### 5.5. Percentile Estimation

#### 5.6. Product Spacing Estimations

#### 5.7. Monte Carlo Simulation

## 6. Empirical Application

#### 6.1. Frequentist Application

#### 6.2. Bayesian Application

## 7. Regression Models

#### 7.1. Quantile Regression Model

#### 7.2. Modal Regression

#### 7.3. Residual Analysis

#### 7.4. Monte Carlo Simulation for Regression Models

#### 7.5. Application of Regression Models

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Parameter | $\mathit{n}$ | ML | MPS | OLS | WLS | AD | CVM | PE | MADS | MALDS |
---|---|---|---|---|---|---|---|---|---|---|

AE | ||||||||||

$\alpha $ | 25 | 0.7609 | 1.1013 | 0.4303 | 0.5079 | 0.5634 | 0.6210 | 0.8969 | 0.1730 | 0.5673 |

50 | 0.8989 | 1.1131 | 0.6865 | 0.7679 | 0.7794 | 0.8387 | 0.9400 | 0.1718 | 0.5865 | |

100 | 0.5186 | 0.6330 | 0.5285 | 0.5408 | 0.5316 | 0.6020 | 0.7364 | 0.3013 | 0.4153 | |

250 | 0.7563 | 0.8212 | 0.6438 | 0.6947 | 0.6821 | 0.6737 | 0.6598 | 0.4516 | 0.5850 | |

350 | 0.8082 | 0.8765 | 0.7720 | 0.8039 | 0.7933 | 0.7969 | 0.6947 | 0.5602 | 0.7547 | |

$\beta $ | 25 | 0.4217 | 0.4674 | 0.3992 | 0.4005 | 0.4065 | 0.4237 | 0.4895 | 0.3323 | 0.4021 |

50 | 0.4294 | 0.4580 | 0.4086 | 0.4158 | 0.4174 | 0.4258 | 0.4584 | 0.2995 | 0.3967 | |

100 | 0.3903 | 0.4039 | 0.3947 | 0.3944 | 0.3926 | 0.4016 | 0.4371 | 0.3582 | 0.3858 | |

250 | 0.4035 | 0.4115 | 0.3938 | 0.3975 | 0.3966 | 0.3974 | 0.4061 | 0.3673 | 0.3940 | |

350 | 0.3949 | 0.4026 | 0.3907 | 0.3944 | 0.3931 | 0.3936 | 0.3904 | 0.3719 | 0.3899 | |

AB | ||||||||||

$\alpha $ | 25 | 0.5584 | 0.6872 | 0.6047 | 0.5382 | 0.5453 | 0.6459 | 0.7676 | 0.6845 | 0.6637 |

50 | 0.5308 | 0.6270 | 0.5159 | 0.5405 | 0.4941 | 0.5491 | 0.9510 | 0.6712 | 0.6083 | |

100 | 0.6628 | 0.6447 | 0.7083 | 0.6909 | 0.6867 | 0.6793 | 0.8618 | 0.5800 | 0.6805 | |

250 | 0.2803 | 0.2719 | 0.3670 | 0.3164 | 0.3256 | 0.3616 | 0.4728 | 0.5443 | 0.4994 | |

350 | 0.2584 | 0.2666 | 0.2306 | 0.2376 | 0.2389 | 0.2336 | 0.4586 | 0.4518 | 0.3332 | |

$\beta $ | 25 | 0.0701 | 0.1000 | 0.0807 | 0.0724 | 0.0686 | 0.0844 | 0.1327 | 0.2182 | 0.1001 |

50 | 0.0442 | 0.0643 | 0.0495 | 0.0435 | 0.0428 | 0.0580 | 0.1059 | 0.1275 | 0.0445 | |

100 | 0.0504 | 0.0530 | 0.0493 | 0.0493 | 0.0490 | 0.0500 | 0.0657 | 0.0640 | 0.0480 | |

250 | 0.0270 | 0.0286 | 0.0352 | 0.0306 | 0.0314 | 0.0358 | 0.0534 | 0.0557 | 0.0356 | |

350 | 0.0226 | 0.0222 | 0.0243 | 0.0176 | 0.0192 | 0.0243 | 0.0520 | 0.0428 | 0.0268 | |

RMSE | ||||||||||

$\alpha $ | 25 | 0.6832 | 0.8824 | 0.6642 | 0.6196 | 0.6373 | 0.7498 | 0.9374 | 0.7249 | 0.7684 |

50 | 0.6291 | 0.7570 | 0.6603 | 0.6831 | 0.5963 | 0.7164 | 1.4860 | 0.7176 | 0.6671 | |

100 | 0.7322 | 0.7492 | 0.7848 | 0.7744 | 0.7611 | 0.7921 | 0.9537 | 0.6576 | 0.7420 | |

250 | 0.3359 | 0.3366 | 0.4614 | 0.3988 | 0.4108 | 0.4615 | 0.5893 | 0.6260 | 0.5625 | |

350 | 0.3129 | 0.3093 | 0.3154 | 0.3086 | 0.3098 | 0.3142 | 0.5602 | 0.5355 | 0.4107 | |

$\beta $ | 25 | 0.0910 | 0.1217 | 0.1029 | 0.0918 | 0.0880 | 0.1174 | 0.1684 | 0.2464 | 0.1214 |

50 | 0.0542 | 0.0782 | 0.0607 | 0.0559 | 0.0493 | 0.0712 | 0.1646 | 0.1592 | 0.0603 | |

100 | 0.0612 | 0.0655 | 0.0627 | 0.0618 | 0.0606 | 0.0652 | 0.0875 | 0.0981 | 0.0604 | |

250 | 0.0337 | 0.0362 | 0.0402 | 0.0364 | 0.0374 | 0.0411 | 0.0679 | 0.0696 | 0.0446 | |

350 | 0.0259 | 0.0259 | 0.0293 | 0.0242 | 0.0249 | 0.0289 | 0.0619 | 0.0560 | 0.0337 |

Parameter | $\mathit{n}$ | ML | MPS | OLS | WLS | AD | CVM | PE | MADS | MALDS |
---|---|---|---|---|---|---|---|---|---|---|

AE | ||||||||||

$\alpha $ | 25 | 7.0765 | 10.3643 | 5.9141 | 5.8055 | 6.6186 | 7.5983 | 4.8574 | 1.2794 | 8.3329 |

50 | 5.0499 | 5.9801 | 4.8062 | 4.7651 | 4.7680 | 5.3690 | 4.1797 | 3.3758 | 5.4587 | |

100 | 4.3862 | 4.8383 | 4.1504 | 4.2629 | 4.2891 | 4.3589 | 3.9500 | 3.6863 | 4.3552 | |

250 | 4.3660 | 4.5560 | 4.2758 | 4.3155 | 4.3307 | 4.3597 | 4.1551 | 3.9716 | 4.4893 | |

350 | 4.3334 | 4.4767 | 4.2076 | 4.2748 | 4.2766 | 4.2668 | 4.2163 | 4.1250 | 4.3294 | |

$\beta $ | 25 | 6.4914 | 7.3170 | 5.9496 | 5.9510 | 6.2163 | 6.5382 | 5.5927 | 3.3139 | 5.9368 |

50 | 6.1885 | 6.6336 | 5.9530 | 6.0059 | 6.0516 | 6.2226 | 5.7082 | 4.6987 | 6.1925 | |

100 | 6.2534 | 6.5278 | 6.0770 | 6.1657 | 6.1849 | 6.2094 | 5.9914 | 5.5851 | 6.2811 | |

250 | 6.1297 | 6.2481 | 6.0714 | 6.1025 | 6.1135 | 6.1240 | 6.0026 | 5.7696 | 6.1201 | |

350 | 6.0608 | 6.1514 | 5.9857 | 6.0232 | 6.0258 | 6.0232 | 5.9824 | 5.8618 | 6.0932 | |

AB | ||||||||||

$\alpha $ | 25 | 3.4127 | 5.9293 | 3.3920 | 3.1570 | 3.4268 | 4.2622 | 2.7449 | 3.2862 | 5.8446 |

50 | 1.8288 | 2.1741 | 2.1320 | 1.9167 | 1.7383 | 2.2757 | 1.7767 | 2.4817 | 2.5227 | |

100 | 1.0012 | 0.9566 | 1.0738 | 1.0249 | 1.0781 | 1.0474 | 1.0521 | 1.5290 | 1.2026 | |

250 | 0.8031 | 0.8054 | 0.8103 | 0.7709 | 0.7570 | 0.7912 | 0.8309 | 1.2029 | 1.0822 | |

350 | 0.6395 | 0.6136 | 0.6138 | 0.6133 | 0.6086 | 0.6041 | 0.6972 | 0.8890 | 0.5945 | |

$\beta $ | 25 | 1.2038 | 1.4981 | 1.3240 | 1.2379 | 1.1823 | 1.3926 | 1.2174 | 2.9029 | 1.2698 |

50 | 0.9340 | 0.9660 | 1.0599 | 0.9933 | 0.9327 | 1.0433 | 1.0666 | 2.1079 | 1.2164 | |

100 | 0.5449 | 0.5436 | 0.5723 | 0.5544 | 0.5715 | 0.5383 | 0.5769 | 0.9975 | 0.6254 | |

250 | 0.4017 | 0.4156 | 0.4049 | 0.4016 | 0.3959 | 0.4026 | 0.4575 | 0.7574 | 0.6456 | |

350 | 0.3707 | 0.3538 | 0.3835 | 0.3678 | 0.3652 | 0.3723 | 0.4190 | 0.5258 | 0.3588 | |

RMSE | ||||||||||

$\alpha $ | 25 | 9.0289 | 16.6588 | 7.7515 | 7.0825 | 8.9366 | 10.7903 | 5.1325 | 3.5862 | 19.9363 |

50 | 3.1101 | 4.1306 | 3.7004 | 2.9429 | 2.7048 | 4.3787 | 2.2720 | 3.1047 | 4.0033 | |

100 | 1.2746 | 1.4424 | 1.3415 | 1.3020 | 1.3645 | 1.3743 | 1.1958 | 2.0602 | 1.7619 | |

250 | 1.0203 | 1.0631 | 1.0172 | 1.0052 | 0.9906 | 1.0217 | 1.0439 | 1.6323 | 1.3097 | |

350 | 0.7575 | 0.7559 | 0.7539 | 0.7476 | 0.7376 | 0.7427 | 0.8050 | 1.2130 | 0.7278 | |

$\beta $ | 25 | 1.5369 | 2.0307 | 1.6441 | 1.5388 | 1.5357 | 1.7984 | 1.4325 | 3.3678 | 1.7998 |

50 | 1.2005 | 1.3372 | 1.3614 | 1.2314 | 1.1733 | 1.3964 | 1.2318 | 2.6988 | 1.5320 | |

100 | 0.6942 | 0.7728 | 0.7270 | 0.6891 | 0.7131 | 0.7296 | 0.6689 | 1.5722 | 0.8371 | |

250 | 0.5388 | 0.5432 | 0.5306 | 0.5343 | 0.5215 | 0.5232 | 0.5916 | 0.9666 | 0.7900 | |

350 | 0.4264 | 0.4122 | 0.4673 | 0.4368 | 0.4343 | 0.4534 | 0.4743 | 0.6624 | 0.4570 |

Model | Parameter | $\mathit{\ell}$ | AIC | DAIC | BIC | AD | CVM | K-S |
---|---|---|---|---|---|---|---|---|

AP | $\alpha =5.0250(0.9841)$ $\beta =8.1856(0.6324)$ | 194.5900 | −385.1756 | 0.0000 | −378.2227 | 0.3670 (0.8806) | 0.0461 (0.8999) | 0.0430 (0.7694) |

AU | $\alpha =2.5208\times {10}^{-14}(0.0828)$ | 0.0000 | 2.0000 | 387.1756 | 5.4765 | 131.0700 (<0.0001) | 28.2090 (<0.0001) | 0.5572 (<0.0001) |

Beta | $\alpha =8.6671(0.8063)$ $\beta =2.2859(0.1962)$ | 191.8700 | −379.7345 | 5.4411 | −372.7816 | 0.8732 (0.4310) | 0.1402 (0.4213) | 0.0650 (0.2647) |

Kumaraswamy | $\alpha =6.6942(0.4546)$ $\beta =2.4355(0.2411)$ | 190.7600 | −377.5820 | 7.5936 | −370.5751 | 1.1438 (0.2899) | 0.1916 (0.2845) | 0.0723 (0.1646) |

UBIII | $\alpha =6.4356(0.5341)$ $\beta =1.5532(0.0695)$ | 192.5000 | −381.0031 | 4.1725 | −374.0501 | 0.7758 (0.4987) | 0.1191 (0.4996) | 0.0535 (0.4997) |

BMOEE | $\alpha =7.6885(1.7248)$ $\beta =9.6771(0.7554)$ | 192.4200 | −380.8355 | 4.3401 | −373.8825 | 0.6848 (0.5715) | 0.0866 (0.6551) | 0.0489 (0.6182) |

UG | $\alpha =1.0457(0.2360)$ $\beta =2.3734(0.3237)$ | 177.0300 | −350.0612 | 35.1144 | −343.1082 | 4.9419 (0.0031) | 0.7829 (0.0080) | 0.1106 (0.0058) |

UW | $\alpha =8.0560(0.8314)$ $\beta =1.6182(0.0791)$ | 192.0200 | −380.0314 | 5.1442 | −373.0785 | 0.8636 (0.4373) | 0.1328 (0.4467) | 0.0557 (0.4486) |

ETL | $\alpha =14.9326(1.3241)$ $\beta =0.8641(0.0718)$ | 192.6800 | −381.3601 | 3.8155 | −374.4072 | 0.6705 (0.5838) | 0.0996 (0.5873) | 0.0520 (0.5370) |

UBXII | $\alpha =10.0760(1.0039)$ $\beta =1.7321(0.0787)$ | 193.5000 | −383.0054 | 2.1702 | −376.0525 | 0.5806 (0.6664) | 0.0887 (0.6437) | 0.0522 (0.5321) |

UISDL | $\alpha =0.3571(0.0134)$ | 54.2900 | −106.5865 | 278.5891 | −103.1101 | 34.4330 (<0.0001) | 20.1010 (<0.0001) | 0.2851 (<0.0001) |

UL | $\alpha =0.2424(0.0112)$ | 97.6400 | −193.2741 | 191.9015 | −189.7976 | 20.1010 (<0.0001) | 4.0961 (<0.0001) | 0.2365 (<0.0001) |

LXL | $\alpha =4.2040(0.2569)$ | 154.6800 | −307.3564 | 77.8192 | −303.8799 | 15.7970 (<0.0001) | 3.0033 (<0.0001) | 0.2010 (<0.0001) |

UPW | $\alpha =500.0000(8.1076\times {10}^{-6})$ $\beta =2.4183(9.9309\times {10}^{-2})$ $\lambda =0.0372(3.5461\times {10}^{-3})$ | 168.2600 | −330.5111 | 54.6645 | −320.0817 | 5.3084 (0.0021) | 0.8375 (0.0059) | 0.1152 (0.0035) |

Parameter | Estimate | SE | SD | 2.50% | 50% | 97.50% | $\widehat{\mathit{R}}$ | Neff |
---|---|---|---|---|---|---|---|---|

$\alpha $ | 5.0600 | 0.0107 | 1.0150 | 3.3760 | 4.9540 | 7.3560 | 1.0010 | 5500 |

$\beta $ | 8.1600 | 0.0066 | 0.6300 | 6.9640 | 8.1490 | 9.4110 | 1.0010 | 6200 |

Parameter | $\mathit{n}$ | AP Quantile Regression | Parameter | $\mathit{n}$ | AP Modal Regression | ||||
---|---|---|---|---|---|---|---|---|---|

AE | AB | RMSE | AE | AB | RMSE | ||||

${\delta}_{0}$ | 50 | 0.7659 | 0.2028 | 0.2533 | ${\delta}_{0}$ | 50 | 0.6495 | 0.5931 | 0.6372 |

150 | 0.7870 | 0.1286 | 0.1586 | 150 | 0.7551 | 0.5240 | 0.5771 | ||

250 | 0.7837 | 0.1041 | 0.1304 | 250 | 0.7015 | 0.4583 | 0.5226 | ||

350 | 0.7953 | 0.0896 | 0.1104 | 350 | 0.7526 | 0.4226 | 0.4880 | ||

450 | 0.7990 | 0.0868 | 0.1071 | 450 | 0.7674 | 0.3745 | 0.4419 | ||

550 | 0.7990 | 0.0681 | 0.0844 | 550 | 0.7668 | 0.3499 | 0.4195 | ||

${\delta}_{1}$ | 50 | 0.4010 | 0.3256 | 0.3983 | ${\delta}_{1}$ | 50 | 0.7202 | 0.6676 | 0.7959 |

150 | 0.3266 | 0.1974 | 0.2407 | 150 | 0.6208 | 0.5630 | 0.7027 | ||

250 | 0.3308 | 0.1737 | 0.2122 | 250 | 0.6470 | 0.5746 | 0.7074 | ||

350 | 0.3119 | 0.1443 | 0.1742 | 350 | 0.5695 | 0.5176 | 0.6518 | ||

450 | 0.3012 | 0.1403 | 0.1711 | 450 | 0.5439 | 0.4813 | 0.6098 | ||

550 | 0.2951 | 0.1044 | 0.1309 | 550 | 0.4965 | 0.4450 | 0.5669 | ||

${\delta}_{2}$ | 50 | 0.6015 | 0.0893 | 0.1157 | ${\delta}_{2}$ | 50 | 0.5921 | 0.3502 | 0.4263 |

150 | 0.6045 | 0.0480 | 0.0614 | 150 | 0.6143 | 0.2171 | 0.2787 | ||

250 | 0.6057 | 0.0381 | 0.0469 | 250 | 0.6090 | 0.1694 | 0.2232 | ||

350 | 0.6006 | 0.0325 | 0.0410 | 350 | 0.6183 | 0.1563 | 0.2020 | ||

450 | 0.6001 | 0.0291 | 0.0371 | 450 | 0.6174 | 0.1259 | 0.1659 | ||

550 | 0.6017 | 0.0272 | 0.0336 | 550 | 0.6187 | 0.1193 | 0.1569 | ||

$\alpha $ | 50 | 1.8184 | 0.7279 | 0.8795 | $\phi $ | 50 | 1.6644 | 0.2465 | 0.2948 |

150 | 1.6469 | 0.4111 | 0.5266 | 150 | 1.5793 | 0.1477 | 0.1879 | ||

250 | 1.5957 | 0.3058 | 0.3971 | 250 | 1.5376 | 0.1026 | 0.1333 | ||

350 | 1.5689 | 0.2526 | 0.3190 | 350 | 1.5289 | 0.0840 | 0.1100 | ||

450 | 1.5586 | 0.2227 | 0.2891 | 450 | 1.5216 | 0.0721 | 0.0931 | ||

550 | 1.5412 | 0.2047 | 0.2602 | 550 | 1.5085 | 0.0693 | 0.0870 |

Parameter | $\mathit{n}$ | AP Quantile Regression | Parameter | $\mathit{n}$ | AP Modal Regression | ||||
---|---|---|---|---|---|---|---|---|---|

AE | AB | RMSE | AE | AB | RMSE | ||||

${\delta}_{0}$ | 50 | 0.1667 | 0.1496 | 0.1906 | ${\delta}_{0}$ | 50 | 0.3746 | 0.3802 | 0.6027 |

150 | 0.1484 | 0.1207 | 0.1502 | 150 | 0.3336 | 0.3336 | 0.5220 | ||

250 | 0.1136 | 0.0907 | 0.1097 | 250 | 0.2376 | 0.2422 | 0.3747 | ||

350 | 0.1171 | 0.0845 | 0.1021 | 350 | 0.2302 | 0.2282 | 0.3570 | ||

450 | 0.1164 | 0.0842 | 0.1028 | 450 | 0.2165 | 0.2085 | 0.3172 | ||

550 | 0.1122 | 0.0714 | 0.0856 | 550 | 0.1841 | 0.1748 | 0.2572 | ||

${\delta}_{1}$ | 50 | 0.4049 | 0.3025 | 0.3523 | ${\delta}_{1}$ | 50 | 0.5759 | 0.5815 | 0.6773 |

150 | 0.3681 | 0.1882 | 0.2312 | 150 | 0.4728 | 0.4831 | 0.5746 | ||

250 | 0.4042 | 0.1654 | 0.2011 | 250 | 0.4892 | 0.4385 | 0.5127 | ||

350 | 0.3862 | 0.1498 | 0.1808 | 350 | 0.4187 | 0.3793 | 0.4540 | ||

450 | 0.3912 | 0.1453 | 0.1771 | 450 | 0.4457 | 0.3684 | 0.4586 | ||

550 | 0.3730 | 0.1047 | 0.1324 | 550 | 0.3974 | 0.3408 | 0.4147 | ||

${\delta}_{2}$ | 50 | 0.7935 | 0.1038 | 0.1363 | ${\delta}_{2}$ | 50 | 0.8970 | 0.3344 | 0.4124 |

150 | 0.8057 | 0.0546 | 0.0699 | 150 | 0.8773 | 0.2046 | 0.2720 | ||

250 | 0.8013 | 0.0426 | 0.0519 | 250 | 0.8651 | 0.1441 | 0.2004 | ||

350 | 0.8008 | 0.0364 | 0.0457 | 350 | 0.8471 | 0.1296 | 0.1734 | ||

450 | 0.7987 | 0.0327 | 0.0414 | 450 | 0.8440 | 0.1052 | 0.1468 | ||

550 | 0.8050 | 0.0326 | 0.0394 | 550 | 0.8339 | 0.1025 | 0.1397 | ||

$\alpha $ | 50 | 1.2087 | 0.3183 | 0.4361 | $\phi $ | 50 | 1.4403 | 0.2164 | 0.2713 |

150 | 1.2667 | 0.2281 | 0.2932 | 150 | 1.3604 | 0.1242 | 0.1589 | ||

250 | 1.2719 | 0.1967 | 0.2448 | 250 | 1.3258 | 0.0870 | 0.1127 | ||

350 | 1.2930 | 0.1702 | 0.2034 | 350 | 1.3211 | 0.0700 | 0.0911 | ||

450 | 1.2871 | 0.1632 | 0.1971 | 450 | 1.3153 | 0.0609 | 0.0785 | ||

550 | 1.2919 | 0.1546 | 0.1845 | 550 | 1.3063 | 0.0588 | 0.0739 |

AP Quantile Regression | AP Modal Regression | ||||||
---|---|---|---|---|---|---|---|

Parameter | Estimate | Standard Error | p-Value | Parameter | Estimate | Standard Error | p-Value |

${\delta}_{0}$ | 1.0119 | 0.1226 | <0.0001 | ${\delta}_{0}$ | 0.8903 | 0.1715 | <0.0001 |

${\delta}_{1}$ | 0.0533 | 0.0912 | 0.5585 | ${\delta}_{1}$ | 0.0921 | 0.1235 | 0.4560 |

${\delta}_{2}$ | 0.2392 | 0.0940 | 0.0110 | ${\delta}_{2}$ | 0.3153 | 0.1559 | 0.0432 |

${\delta}_{3}$ | 0.0169 | 0.0049 | 0.0006 | ${\delta}_{3}$ | 0.0253 | 0.0082 | 0.0020 |

$\alpha $ | 5.6100 | 1.1128 | <0.0001 | $\phi $ | 8.4244 | 0.6471 | <0.0001 |

$\ell =201.1400$ | $\ell =199.7300$ | ||||||

$\mathrm{AIC}=-392.2835$ | $\mathrm{AIC}=-389.4540$ | ||||||

$\mathrm{BIC}=-374.9012$ | $\mathrm{BIC}=-372.0717$ |

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## Share and Cite

**MDPI and ACS Style**

Nasiru, S.; Abubakari, A.G.; Chesneau, C.
The Arctan Power Distribution: Properties, Quantile and Modal Regressions with Applications to Biomedical Data. *Math. Comput. Appl.* **2023**, *28*, 25.
https://doi.org/10.3390/mca28010025

**AMA Style**

Nasiru S, Abubakari AG, Chesneau C.
The Arctan Power Distribution: Properties, Quantile and Modal Regressions with Applications to Biomedical Data. *Mathematical and Computational Applications*. 2023; 28(1):25.
https://doi.org/10.3390/mca28010025

**Chicago/Turabian Style**

Nasiru, Suleman, Abdul Ghaniyyu Abubakari, and Christophe Chesneau.
2023. "The Arctan Power Distribution: Properties, Quantile and Modal Regressions with Applications to Biomedical Data" *Mathematical and Computational Applications* 28, no. 1: 25.
https://doi.org/10.3390/mca28010025