# Extended Generalized Sinh-Normal Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Extended Sinh-Normal Distribution

## 3. Moments of an Extended Sinh-Normal Random Variable

## 4. Extended Generalized Sinh-Normal Distribution

#### 4.1. Stochastic Representation

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**1.**

**Proof.**

**Definition**

**2.**

**Proposition**

**2.**

**Proof.**

#### Location-Scale Extension

#### 4.2. Moments of an Extended Generalized Sinh-Normal Random Variable

#### 4.3. Extended Generalized Birnbaum–Saunders Distribution

## 5. Extended Sinh-Normal Regression Model

- ${\beta}_{i}=exp\left({x}_{i}^{\prime}\mathbf{\theta}\right),$ for $i=1,2,\dots ,n,$ with ${\mathbf{\theta}}^{\prime}=({\theta}_{1},{\theta}_{2},...,{\theta}_{p}),$ being a p-dimensional vector of unknown parameters.
- The shape, bimodality, and skew parameters do not consider ${x}_{i};$ i.e., ${\alpha}_{i}=\alpha ,{\gamma}_{i}=\gamma ,\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{i}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\lambda \phantom{\rule{3.33333pt}{0ex}}\mathrm{for}\phantom{\rule{3.33333pt}{0ex}}i=1,2,\dots ,n.$

## 6. Simulation Study

## 7. Numerical Illustrations

#### 7.1. Illustration 1

#### 7.2. Illustration 2

## 8. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Distribution (

**a**) $ESHN(2.75,0,1,\gamma )$ for $\gamma =3.5$ (solid line), $\gamma =2.5$ (dashed line), $\gamma =1.5$ (dotted line) y $\gamma =0$ (dash-dotted line), (

**b**) $ESHN(1.75,0,1,\gamma )$ for $\gamma =3.5$ (solid line), $\gamma =2.5$ (dashed line), $\gamma =1.5$ (dotted line) and $\gamma =0$ (dash-dotted line) and (

**c**) $ESHN(0.75,0,1,\gamma )$ for $\gamma =3.5$ (solid line), $\gamma =2.5$ (dashed line), $\gamma =1.5$ (dotted line) and $\gamma =0$ (dash-dotted line).

**Figure 2.**EGSHN distribution (

**a**) $ESHN(0.75,0,1,3.5,-0.75)$ (solid line), $ESHN(0.75,0,1,2.5,-0.5)$ (dashed line), $ESHN(0.75,0,1,0.5,1.5)$ (dotted line) and $ESHN(0.75,0,1,1.5,0.75)$ (dash-dotted line), (

**b**) $ESHN(1.5,0,1,3.5,-2.5)$ (solid line), $ESHN(1.5,0,1,2.5,-1.5)$ (dashed line), $ESHN(1.5,0,1,1.5,2.5)$ (dotted line) and $ESHN(1.5,0,1,0.5,1.5)$ (dash-dotted line) and (

**c**) $ESHN(2.5,0,1,3.5,-0.25)$ (solid line), $ESHN(2.5,0,1,3.5,-0.75)$ (dashed line), $ESHN(2.5,0,1,2.5,0.75)$ (dotted line) and $ESHN(2.5,0,1,0.75,0.25)$ (dash-dotted line).

**Figure 3.**Empirical sd, relative bias, and $\sqrt{MSE}$ for the estimators of the $EGSHN(\alpha ,2.25,0.75,1.5,1)$ model parameters with sample sizes of 30, 60, 90, 120 and 500.

**Figure 4.**Empirical sd, relative bias, and $\sqrt{MSE}$ for the estimators of the $EGSHN(1.75,2.25,0.75,\gamma ,1)$ model parameters with sample sizes of 30, 60, 90, 120 and 500.

**Figure 5.**Empirical sd, relative bias, and $\sqrt{MSE}$ for the estimators of the $EGSHN$ $(1.75,2.25,0.75,2.0,\lambda )$ model parameters with sample sizes of 30, 60, 90, 120 and 500.

**Figure 6.**Normal probability plots for $rM{T}_{i}$ with envelopes of Q-qplots for the scaled residuals, from the fitted models. (

**a**) ESHN and (

**b**) EGSHN.

**Figure 7.**Influence measures for the EGSHN model (

**a**) Cook’s distance, (

**b**) $rM{T}_{i}$, (

**c**) envelope picture of EGSHN model.

**Figure 8.**(

**a**) Histogram of the variable (amount of DNA in cancer cells) for the EGBS (solid line), EBS (dashed line), BLSN (dotted line), and BSSN (dash-dotted line) adjusted distributions; and (

**b**) empirical cumulative distribution (solid line) and for the EGBS (dashed line), EBS (dotted line), and BLSN (dash-dotted line) models.

**Table 1.**Empirical sd, relative bias, and $\sqrt{MSE}$ for the $EGSHN(\alpha ,2.25,0.75,1.5,1)$ model.

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\lambda}}$ | ${\widehat{\mathit{\beta}}}_{0}$ | ${\widehat{\mathit{\beta}}}_{1}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | n | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ |

30 | 0.1600 | 0.0903 | 0.1737 | 3.4599 | 1.1505 | 3.8661 | 0.6997 | 0.0677 | 0.7029 | 0.3332 | 0.0386 | 0.3443 | 0.4105 | 0.0177 | 0.4107 | |

60 | 0.1339 | 0.0502 | 0.1390 | 2.5812 | 0.7354 | 2.8067 | 0.4756 | 0.0380 | 0.4770 | 0.2737 | 0.0226 | 0.2784 | 0.2818 | 0.0145 | 0.2819 | |

0.75 | 90 | 0.1194 | 0.0311 | 0.1216 | 1.9235 | 0.5013 | 2.0651 | 0.4127 | 0.0216 | 0.4132 | 0.2396 | 0.0117 | 0.241 | 0.2252 | 0.0046 | 0.2252 |

120 | 0.1087 | 0.0187 | 0.1096 | 1.2017 | 0.3818 | 1.3311 | 0.3681 | 0.0028 | 0.3681 | 0.2147 | 0.0040 | 0.2148 | 0.1897 | 0.0034 | 0.1897 | |

500 | 0.0541 | 0.0038 | 0.0542 | 0.3686 | 0.0705 | 0.3834 | 0.1759 | 0.0002 | 0.1759 | 0.1069 | 0.0017 | 0.1069 | 0.0913 | 0.0026 | 0.0913 | |

30 | 0.4831 | 0.0640 | 0.4958 | 6.4445 | 1.1316 | 6.6637 | 0.5962 | 0.0201 | 0.5965 | 0.5550 | 0.0531 | 0.5677 | 0.6102 | 0.0124 | 0.6102 | |

60 | 0.4162 | 0.0263 | 0.4187 | 1.9279 | 0.5686 | 2.1079 | 0.5113 | 0.0198 | 0.5116 | 0.4404 | 0.0211 | 0.4429 | 0.4255 | 0.0093 | 0.4255 | |

1.75 | 90 | 0.3661 | 0.0135 | 0.3669 | 1.2541 | 0.3858 | 1.3811 | 0.4518 | 0.0292 | 0.4527 | 0.3657 | 0.0086 | 0.3662 | 0.3324 | 0.0137 | 0.3325 |

120 | 0.3354 | 0.0063 | 0.3356 | 0.8840 | 0.2594 | 0.9657 | 0.4098 | 0.0267 | 0.4107 | 0.3303 | 0.0072 | 0.3307 | 0.2839 | 0.0040 | 0.2839 | |

500 | 0.1476 | 0.0012 | 0.1476 | 0.3285 | 0.0486 | 0.3364 | 0.1646 | 0.0101 | 0.1649 | 0.1568 | 0.0001 | 0.1568 | 0.1379 | 0.0036 | 0.1379 | |

30 | 0.8355 | 0.0578 | 0.8504 | 5.0777 | 1.0509 | 5.3163 | 0.6025 | 0.0095 | 0.6025 | 0.6021 | 0.0555 | 0.6149 | 0.6722 | 0.0149 | 0.6722 | |

60 | 0.7030 | 0.0225 | 0.7056 | 2.2156 | 0.5590 | 2.3687 | 0.5214 | 0.0280 | 0.5221 | 0.4612 | 0.0257 | 0.4648 | 0.4591 | 0.0102 | 0.4591 | |

2.75 | 90 | 0.6090 | 0.0079 | 0.6094 | 1.2718 | 0.3672 | 1.3858 | 0.4443 | 0.0301 | 0.4452 | 0.3931 | 0.0139 | 0.3943 | 0.3725 | 0.0151 | 0.3727 |

120 | 0.5096 | 0.0104 | 0.5104 | 0.9270 | 0.2499 | 0.9998 | 0.3758 | 0.0157 | 0.3761 | 0.3296 | 0.0127 | 0.3308 | 0.3076 | 0.0022 | 0.3075 | |

500 | 0.2182 | 0.0033 | 0.2184 | 0.3371 | 0.0563 | 0.3475 | 0.1417 | 0.0027 | 0.1417 | 0.1525 | 0.0022 | 0.1525 | 0.1496 | 0.0000 | 0.1496 |

**Table 2.**Empirical sd, relative bias, and $\sqrt{MSE}$ for the $EGSHN(1.75,2.25,0.75,\gamma ,1)$ model.

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\lambda}}$ | ${\widehat{\mathit{\beta}}}_{0}$ | ${\widehat{\mathit{\beta}}}_{1}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\gamma}$ | n | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ |

30 | 0.5060 | 0.0784 | 0.5242 | 2.6997 | 1.3552 | 3.0205 | 0.6351 | 0.0473 | 0.6368 | 0.5845 | 0.0553 | 0.5975 | 0.6630 | 0.0204 | 0.6631 | |

60 | 0.4768 | 0.0359 | 0.4809 | 1.4215 | 0.6769 | 1.5743 | 0.6234 | 0.0014 | 0.6233 | 0.4910 | 0.0272 | 0.4948 | 0.4506 | 0.0130 | 0.4507 | |

1.0 | 90 | 0.4368 | 0.0159 | 0.4376 | 0.7941 | 0.4384 | 0.9071 | 0.5594 | 0.0290 | 0.5601 | 0.4321 | 0.0130 | 0.4330 | 0.3520 | 0.0017 | 0.3520 |

120 | 0.3930 | 0.0121 | 0.3935 | 0.5962 | 0.2952 | 0.6652 | 0.5107 | 0.0281 | 0.5115 | 0.3900 | 0.0113 | 0.3908 | 0.3088 | 0.0100 | 0.3088 | |

500 | 0.1957 | 0.0000 | 0.1957 | 0.2305 | 0.0703 | 0.2409 | 0.2402 | 0.0121 | 0.2404 | 0.1977 | 0.0005 | 0.1977 | 0.1493 | 0.0026 | 0.1493 | |

30 | 0.4612 | 0.0446 | 0.4677 | 6.0222 | 0.7279 | 6.2906 | 0.5485 | 0.0185 | 0.5487 | 0.5092 | 0.0417 | 0.5178 | 0.5756 | 0.0272 | 0.5759 | |

60 | 0.3695 | 0.0123 | 0.3701 | 4.8084 | 0.6148 | 5.0476 | 0.4371 | 0.0360 | 0.4386 | 0.3835 | 0.0093 | 0.3841 | 0.3859 | 0.0084 | 0.3859 | |

2.5 | 90 | 0.3035 | 0.0089 | 0.3039 | 2.7667 | 0.3974 | 2.9394 | 0.3429 | 0.0270 | 0.3439 | 0.3177 | 0.0083 | 0.3182 | 0.3121 | 0.0062 | 0.3121 |

120 | 0.2655 | 0.0026 | 0.2655 | 1.9695 | 0.2684 | 2.0805 | 0.3056 | 0.0274 | 0.3068 | 0.2768 | 0.0029 | 0.2768 | 0.2694 | 0.0026 | 0.2690 | |

500 | 0.1129 | 0.0007 | 0.1129 | 0.5670 | 0.0575 | 0.5849 | 0.1213 | 0.0049 | 0.1214 | 0.1257 | 0.0008 | 0.1257 | 0.1270 | 0.0014 | 0.1270 | |

30 | 0.4246 | 0.0245 | 0.4267 | 13.3364 | 0.6570 | 13.5521 | 0.5613 | 0.0693 | 0.5655 | 0.4786 | 0.0269 | 0.4824 | 0.5404 | 0.0179 | 0.5406 | |

60 | 0.3298 | 0.0048 | 0.3298 | 11.4564 | 0.5569 | 11.7093 | 0.4094 | 0.0631 | 0.4142 | 0.3450 | 0.0060 | 0.3452 | 0.3628 | 0.0054 | 0.3628 | |

4.0 | 90 | 0.2629 | 0.0015 | 0.2629 | 5.9366 | 0.4821 | 6.2414 | 0.2938 | 0.0363 | 0.2960 | 0.2792 | 0.0002 | 0.2792 | 0.2922 | 0.0103 | 0.2922 |

120 | 0.2210 | 0.0018 | 0.2210 | 5.6103 | 0.3773 | 5.8091 | 0.2506 | 0.0218 | 0.2515 | 0.2328 | 0.0005 | 0.2328 | 0.2498 | 0.0027 | 0.2498 | |

500 | 0.0982 | 0.0007 | 0.0982 | 1.0651 | 0.0704 | 1.1016 | 0.1075 | 0.0057 | 0.1076 | 0.1111 | 0.0001 | 0.1111 | 0.1218 | 0.0018 | 0.1218 |

**Table 3.**Empirical sd, relative bias, and $\sqrt{MSE}$ for the $EGSHN(1.75,2.25,0.75,2.0,\lambda )$ model.

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\lambda}}$ | ${\widehat{\mathit{\beta}}}_{0}$ | ${\widehat{\mathit{\beta}}}_{1}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\lambda}$ | n | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ | sd | RB | $\sqrt{\mathit{MSE}}$ |

30 | 0.3354 | 0.0083 | 0.3357 | 5.5409 | 0.7808 | 5.7563 | 0.3244 | 0.0541 | 0.3255 | 0.4508 | 0.0048 | 0.4509 | 0.6493 | 0.0215 | 0.6494 | |

60 | 0.2278 | 0.0042 | 0.2279 | 3.0014 | 0.5044 | 3.1661 | 0.2077 | 0.0389 | 0.2086 | 0.2991 | 0.0020 | 0.2991 | 0.4186 | 0.0087 | 0.4186 | |

0.50 | 90 | 0.1779 | 0.0055 | 0.1782 | 2.2964 | 0.3269 | 2.3874 | 0.1637 | 0.0189 | 0.1640 | 0.2386 | 0.0014 | 0.2386 | 0.3376 | 0.0011 | 0.3376 |

120 | 0.1570 | 0.0008 | 0.1570 | 1.2968 | 0.2010 | 1.3575 | 0.1412 | 0.0246 | 0.1417 | 0.2094 | 0.0011 | 0.2094 | 0.2892 | 0.0046 | 0.2890 | |

500 | 0.0711 | 0.0018 | 0.0711 | 0.4458 | 0.0465 | 0.4554 | 0.0639 | 0.0035 | 0.0639 | 0.0955 | 0.0000 | 0.0955 | 0.1374 | 0.0053 | 0.1374 | |

30 | 0.4651 | 0.1093 | 0.5029 | 4.9920 | 0.8175 | 5.2525 | 0.9968 | 0.0428 | 0.9988 | 0.5470 | 0.0949 | 0.5872 | 0.5720 | 0.0377 | 0.5727 | |

60 | 0.4308 | 0.0552 | 0.4414 | 3.8817 | 0.6443 | 4.0896 | 0.7276 | 0.0189 | 0.7276 | 0.4703 | 0.0479 | 0.4824 | 0.3853 | 0.0162 | 0.3855 | |

1.5 | 90 | 0.4041 | 0.0385 | 0.4096 | 2.7730 | 0.4517 | 2.9162 | 0.6860 | 0.0133 | 0.6860 | 0.4268 | 0.0335 | 0.4333 | 0.3179 | 0.0066 | 0.3179 |

120 | 0.3746 | 0.0211 | 0.3764 | 1.5575 | 0.3195 | 1.6834 | 0.6473 | 0.0097 | 0.6475 | 0.3894 | 0.0179 | 0.3915 | 0.2758 | 0.0123 | 0.2760 | |

500 | 0.2312 | 0.0024 | 0.2312 | 0.5188 | 0.0785 | 0.5420 | 0.3727 | 0.0013 | 0.3738 | 0.2313 | 0.0004 | 0.2313 | 0.1314 | 0.0037 | 0.1314 | |

30 | 0.3930 | 0.1271 | 0.4515 | 3.4510 | 0.3915 | 3.5384 | 17.0967 | 0.1390 | 17.0977 | 0.4872 | 0.1168 | 0.5535 | 0.5789 | 0.0631 | 0.5808 | |

60 | 0.3635 | 0.0677 | 0.3823 | 3.1596 | 0.3342 | 3.2292 | 8.0921 | 0.1382 | 8.0979 | 0.4215 | 0.0661 | 0.4469 | 0.3792 | 0.0234 | 0.3796 | |

3.0 | 90 | 0.3421 | 0.0446 | 0.3508 | 3.0435 | 0.2479 | 3.0834 | 2.4923 | 0.1378 | 2.5170 | 0.4013 | 0.0493 | 0.4163 | 0.3132 | 0.0092 | 0.3133 |

120 | 0.3237 | 0.0240 | 0.3264 | 2.0577 | 0.2166 | 2.1026 | 2.0773 | 0.1180 | 2.1181 | 0.3781 | 0.0313 | 0.3846 | 0.2592 | 0.0039 | 0.2592 | |

500 | 0.2590 | 0.0113 | 0.2597 | 1.1857 | 0.1159 | 1.208 | 1.6695 | 0.1095 | 1.7197 | 0.2996 | 0.0019 | 0.2996 | 0.1298 | 0.0018 | 0.1290 |

Estimator | SHN | ESHN | EGSHN |
---|---|---|---|

$\widehat{\alpha}$ | 245.9799 | 66.9952 | 7.9489 |

(230.85) | (48.8845) | (2.6274) | |

${\widehat{\beta}}_{0}$ | 9.2750 | 9.3422 | 9.3456 |

(0.1595) | (0.1666) | (0.1616) | |

${\widehat{\beta}}_{1}$ | −0.4217 | −0.4077 | −0.4144 |

(0.0190) | (0.0165) | (0.0189) | |

$\widehat{\sigma}$ | 0.3572 | 0.4306 | 0.7539 |

(0.0536) | (0.0585) | (0.0903) | |

$\widehat{\gamma}$ | 0.3070 | 6.1671 | |

(0.1166) | (3.3568) | ||

$\widehat{\lambda}$ | −0.6493 | ||

(0.1965) | |||

AIC | 89.59 | 74.42 | 73.1924 |

AICC | 93.30 | 78.89 | 78.58 |

Observation | $\widehat{\mathit{\alpha}}$ | ${\widehat{\mathit{\beta}}}_{0}$ | ${\widehat{\mathit{\beta}}}_{1}$ | $\widehat{\mathit{\sigma}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\lambda}}$ |
---|---|---|---|---|---|---|

8 | 0.0752 | 1.0751 | 3.6291 | 4.4136 | 0.2496 | 9.5977 |

23 | 0.1942 | 0.0200 | 0.2091 | 1.2090 | 0.0877 | 2.5542 |

34 | 0.2357 | 1.5518 | 4.3536 | 1.0540 | 0.3764 | 7.9936 |

8, 23 | 0.0839 | 0.8594 | 2.8812 | 5.7076 | 0.3233 | 13.8282 |

8, 34 | 0.0253 | 0.6924 | 2.0139 | 3.4098 | 0.1873 | 10.0372 |

23, 34 | 0.0731 | 0.9189 | 3.0378 | 0.6211 | 0.2350 | 8.8319 |

8, 23, 34 | 0.0326 | 0.6332 | 1.7068 | 5.2954 | 0.2438 | 14.2785 |

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

3.636 | 1.432 | 0.452 | 0.865 |

Estimators | BS | BSSN | BLSN | EBS | EGBS |
---|---|---|---|---|---|

$\widehat{\alpha}$ | (0.3145) | 0.5254 | 1.3564 | 0.2033 | 0.2136 |

(0.0140) | (0.0263) | (0.0169) | (0.0066) | (0.0082) | |

$\widehat{\beta}$ | 3.5194 | 2.3042 | 0.2066 | 3.7995 | 4.0200 |

(0.0698) | (0.032) | (0.0070) | (0.0551) | (0.0643) | |

${\widehat{\gamma}}_{1}$ | 3.9845 | 4.4899 | 5.6760 | ||

(1.2161) | (1.7541) | (2.5460) | |||

$\widehat{\lambda}$ | 7.7814 | −0.2874 | −0.3724 | ||

(1.2943) | (0.0677) | (0.0701) | |||

AIC | 745.58 | 698.51 | 671.95 | 668.41 | 637.8484 |

AICC | 747.68 | 700.68 | 674.20 | 670.58 | 640.09 |

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**MDPI and ACS Style**

Martínez-Flórez, G.; Elal-Olivero, D.; Barrera-Causil, C.
Extended Generalized Sinh-Normal Distribution. *Mathematics* **2021**, *9*, 2793.
https://doi.org/10.3390/math9212793

**AMA Style**

Martínez-Flórez G, Elal-Olivero D, Barrera-Causil C.
Extended Generalized Sinh-Normal Distribution. *Mathematics*. 2021; 9(21):2793.
https://doi.org/10.3390/math9212793

**Chicago/Turabian Style**

Martínez-Flórez, Guillermo, David Elal-Olivero, and Carlos Barrera-Causil.
2021. "Extended Generalized Sinh-Normal Distribution" *Mathematics* 9, no. 21: 2793.
https://doi.org/10.3390/math9212793