# The Exponentiated Truncated Inverse Weibull-Generated Family of Distributions with Applications

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## Abstract

**:**

## 1. Introduction

## 2. The ETIW-G Family

#### 2.1. Probability Functions

#### 2.2. Some Special Members of the ETIW-G Family

#### 2.3. The ETIWIW Distribution

## 3. Mathematical Investigations

#### 3.1. Stochastic Ordering Results

**Proposition**

**1.**

- Let${G}_{1}(x;\xi )$and${G}_{2}(x;\xi )$be two baseline cdfs, and${F}_{G}(x;a,b,\xi )$be defined as (2). Then, if${G}_{2}(x;\xi )\ge {G}_{1}(x;\xi )$, for any$x\in \mathbb{R}$, we have$${F}_{{G}_{1}}(x;a,b,\xi )\le {F}_{{G}_{2}}(x;a,b,\xi ).$$
- For any${b}_{*}\ge b>0$and${a}_{*}\ge a>0$, and any$x\in \mathbb{R}$, we have$$F(x;{a}_{*},b,\xi )\le F(x;a,{b}_{*},\xi ).$$

**Proof.**

- The desired inequality follows from the fact that ${F}_{G}(x;a,b,\xi )={F}_{\diamond}(G(x;\xi );a,b)$, where ${F}_{\diamond}(x;a,b)={\left(1-{e}^{1-{(1-x)}^{-b}}\right)}^{a}$, $x\in (0,1)$, is an increasing function with respect to x (as the cdf of the exponentiated type II truncated inverse Weibull distribution over $(0,1)$).
- Since $G(x;\xi )\in (0,1)$ (excluding the limit cases), ${b}_{*}\ge b>0$ and ${a}_{*}\ge a>0$, we have the following chain of equivalences:$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & {(1-G(x;\xi ))}^{{b}_{*}}\le {(1-G(x;\xi ))}^{b}\phantom{\rule{1.em}{0ex}}\iff \phantom{\rule{1.em}{0ex}}{(1-G(x;\xi ))}^{-b}\le {(1-G(x;\xi ))}^{-{b}_{*}}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \iff \phantom{\rule{1.em}{0ex}}1-{(1-G(x;\xi ))}^{-{b}_{*}}\le 1-{(1-G(x;\xi ))}^{-b}\phantom{\rule{1.em}{0ex}}\iff \phantom{\rule{1.em}{0ex}}{e}^{1-{(1-G(x;\xi ))}^{-{b}_{*}}}\le {e}^{1-{(1-G(x;\xi ))}^{-b}}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \iff \phantom{\rule{1.em}{0ex}}1-{e}^{1-{(1-G(x;\xi ))}^{-b}}\le 1-{e}^{1-{(1-G(x;\xi ))}^{-{b}_{*}}}\phantom{\rule{1.em}{0ex}}\iff \hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \iff \phantom{\rule{1.em}{0ex}}{\left(1-{e}^{1-{(1-G(x;\xi ))}^{-b}}\right)}^{{a}_{*}}\le {\left(1-{e}^{1-{(1-G(x;\xi ))}^{-b}}\right)}^{a}\le {\left(1-{e}^{1-{(1-G(x;\xi ))}^{-{b}_{*}}}\right)}^{a},\hfill \end{array}$$

#### 3.2. Uni/multimodality Analysis

#### 3.3. Tractable Series Expansions

**Proposition**

**2.**

**Proof.**

#### 3.4. Probability Weighted Moments

#### 3.5. Raw and Central Moments, with Applications

#### 3.6. Order Statistics

**Proposition**

**3.**

**Proof.**

#### 3.7. Maximum Likelihood Method

## 4. Numerical Studies

#### 4.1. Simulation Work

#### 4.2. Application to the Rainfall Data Set

#### 4.3. Application to the Sum of Skin Folds Data Set

#### 4.4. Application to the Completed Passes of Drew Brees Data Set

## 5. Concluding Remarks and Perspectives

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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ETIW-G | Baseline Distribution | $\mathit{\xi}$ | Support | $\mathit{G}(\mathit{x};\mathit{\xi})$ | $\mathit{F}(\mathit{x};\mathit{a},\mathit{b},\mathit{\xi})$ |
---|---|---|---|---|---|

ETIWU | Uniform | $\left(\theta \right)$ | $(0,\theta )$ | $x/\theta $ | ${\left(1-{e}^{1-{(1-x/\theta )}^{-b}}\right)}^{a}$ |

ETIWP | Power | $\left(a\right)$ | $(0,1)$ | ${x}^{\alpha}$ | ${\left(1-{e}^{1-{(1-{x}^{\alpha})}^{-b}}\right)}^{a}$ |

ETIWD | Dagum | $(\alpha ,\beta ,\gamma )$ | $(0,+\infty )$ | ${\left(1+{\left(x/\alpha \right)}^{-\beta}\right)}^{-\gamma}$ | ${\left(1-{e}^{1-{\left[1-{\left(1+{\left(x/\alpha \right)}^{-\beta}\right)}^{-\gamma}\right]}^{-b}}\right)}^{a}$ |

ETIWLi | Lindley | $\left(\theta \right)$ | $(0,+\infty )$ | $1-(1+\theta +\theta x){e}^{-\theta x}/(1+\theta )$ | ${\left(1-{e}^{1-{(1+\theta )}^{b}{(1+\theta +\theta x)}^{-b}{e}^{b\theta x}}\right)}^{a}$ |

ETIWHC | Half Cauchy | $\left(\alpha \right)$ | $(0,+\infty )$ | $(2/\pi )arctan(x/\alpha )$ | ${\left(1-{e}^{1-{(1-(2/\pi )arctan(x/\alpha ))}^{-b}}\right)}^{a}$ |

ETIWG | Gamma | $(\alpha ,\beta )$ | $(0,+\infty )$ | $\gamma (\alpha ,\beta x)/\mathsf{\Gamma}\left(\alpha \right)$ | ${\left(1-{e}^{1-{(1-\gamma (\alpha ,\beta x)/\mathsf{\Gamma}\left(\alpha \right))}^{-b}}\right)}^{a}$ |

ETIWLo | Logistic | $(\alpha ,\beta )$ | $\mathbb{R}$ | ${\left(1+{e}^{-(x-\alpha )/\beta}\right)}^{-1}$ | ${\left(1-{e}^{1-{\left[1-{(1+{e}^{-(x-\alpha )/\beta})}^{-1}\right]}^{-b}}\right)}^{a}$ |

ETIWN | Normal | $(\mu ,\sigma )$ | $\mathbb{R}$ | $\Phi (x;\mu ,\sigma )$ | ${\left(1-{e}^{1-{(1-\Phi (x;\mu ,\sigma ))}^{-b}}\right)}^{a}$ |

n | Set1 | Set2 | Set3 | |||
---|---|---|---|---|---|---|

MLE | MSE | MLE | MSE | MLE | MSE | |

50 | 0.747 | 0.159 | 1.028 | 0.734 | 1.377 | 1.031 |

0.507 | 0.065 | 1.729 | 1.568 | 1.704 | 1.279 | |

0.729 | 0.436 | 0.768 | 0.128 | 0.678 | 0.059 | |

0.707 | 0.093 | 2.711 | 2.992 | 3.322 | 2.939 | |

100 | 0.737 | 0.153 | 1.318 | 0.599 | 1.463 | 0.825 |

0.502 | 0.027 | 1.522 | 0.631 | 1.393 | 0.449 | |

0.662 | 0.227 | 0.698 | 0.073 | 0.641 | 0.034 | |

0.662 | 0.046 | 2.227 | 0.896 | 3.204 | 2.127 | |

200 | 0.718 | 0.124 | 1.31 | 0.29 | 1.6 | 0.501 |

0.487 | 0.014 | 1.241 | 0.361 | 1.229 | 0.268 | |

0.66 | 0.22 | 0.636 | 0.036 | 0.605 | 0.019 | |

0.662 | 0.036 | 2.204 | 0.827 | 3.075 | 1.481 | |

500 | 0.786 | 0.12 | 1.394 | 0.208 | 1.644 | 0.344 |

0.475 | 0.0063 | 1.184 | 0.185 | 1.209 | 0.149 | |

0.475 | 0.09 | 0.611 | 0.02 | 0.598 | 0.013 | |

0.645 | 0.026 | 2.168 | 0.55 | 2.943 | 1.037 | |

1000 | 0.761 | 0.087 | 1.394 | 0.125 | 1.673 | 0.241 |

0.464 | 0.0042 | 1.15 | 0.165 | 1.335 | 0.125 | |

0.449 | 0.03 | 0.601 | 0.015 | 0.584 | 0.0084 | |

0.651 | 0.025 | 2.152 | 0.499 | 2.948 | 0.975 |

n | Set4 | Set5 | Set6 | |||
---|---|---|---|---|---|---|

MLE | MSE | MLE | MSE | MLE | MSE | |

50 | 1.016 | 0.934 | 1.138 | 0.087 | 1.272 | 1.642 |

1.299 | 2.379 | 0.438 | 0.054 | 1.76 | 3.227 | |

1.604 | 4.595 | 0.926 | 0.279 | 1.461 | 2.553 | |

1.2 | 0.391 | 0.93 | 0.413 | 1.309 | 0.637 | |

100 | 0.968 | 0.48 | 1.152 | 0.06 | 1.304 | 1.152 |

1.034 | 0.574 | 0.445 | 0.03 | 1.359 | 0.719 | |

1.146 | 1.367 | 0.927 | 0.244 | 1.097 | 0.919 | |

1.05 | 0.175 | 0.813 | 0.173 | 1.183 | 0.29 | |

200 | 0.985 | 0.398 | 1.249 | 0.042 | 1.477 | 1.021 |

0.908 | 0.184 | 0.455 | 0.019 | 1.221 | 0.45 | |

0.905 | 0.611 | 0.862 | 0.18 | 0.88 | 0.498 | |

1.01 | 0.108 | 0.756 | 0.1 | 1.102 | 0.196 | |

500 | 0.978 | 0.309 | 1.238 | 0.018 | 1.485 | 0.352 |

0.734 | 0.035 | 0.463 | 0.0075 | 0.994 | 0.136 | |

0.65 | 0.119 | 0.852 | 0.162 | 0.672 | 0.092 | |

0.937 | 0.074 | 0.713 | 0.054 | 1.096 | 0.131 | |

1000 | 0.907 | 0.158 | 1.229 | 0.011 | 1.511 | 0.109 |

0.686 | 0.024 | 0.465 | 0.0061 | 0.985 | 0.103 | |

0.57 | 0.033 | 0.815 | 0.13 | 0.617 | 0.046 | |

0.854 | 0.065 | 0.711 | 0.053 | 1.049 | 0.093 |

Model | a | b | $\mathit{\mu}$ | $\mathit{\theta}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|

ETIWIW | 0.4443 | 477.2912 | 118.7575 | 0.6907 | - | - |

(0.1084) | (8.6456) | (5.4174) | (0.0392) | - | - | |

TIITFIW | - | 40.7759 | 62.1370 | 0.6356 | - | - |

- | (7.9297) | (1.14315) | (0.0982) | - | - | |

LGFr | 0.2832 | 23.0128 | 21.4924 | 1.3365 | - | - |

(0.0413) | (7.8511) | (2.2458) | (1.1915) | - | - | |

GOFr | - | - | 13.0778 | 4.6917 | 0.1739 | 1.1748 |

- | - | (9.1451) | (3.4160) | (0.8502) | (0.3630) | |

KwFr | 5.2239 | 554.6118 | 9.8825 | 0.5019 | - | - |

(36.8518) | (22.0773) | (38.8463) | (0.0927) | - | - | |

BFr | - | - | 71.1502 | 259.1749 | 74.9748 | 0.1685 |

- | - | (12.1141) | (30.5647) | (1.0465) | (3.1490) | |

EFr | - | - | 1.6108 | 1.9444 | 7.9293 | - |

- | - | (67.4174) | (1.0673) | (1.0274) | - |

**Table 5.**Values of $-\widehat{\ell}$, AIC, W*, A*, K-S and the related p-value for the rainfall data set.

Model | $-\widehat{\mathit{\ell}}$ | AIC | W* | A* | K-S | p-Value |
---|---|---|---|---|---|---|

ETIWIW | 80.8218 | 169.6438 | 0.0540 | 0.4239 | 0.1185 | 0.6276 |

TIITFIW | 82.9131 | 171.8263 | 0.0905 | 0.6681 | 0.1274 | 0.5340 |

LGFr | 88.9168 | 185.8337 | 0.2372 | 1.5496 | 0.1820 | 0.1412 |

GOFr | 87.8787 | 183.7575 | 0.2201 | 1.4468 | 0.1353 | 0.4562 |

KwFr | 86.1506 | 180.3014 | 0.1710 | 1.1615 | 0.1218 | 0.5928 |

BFr | 91.2658 | 190.5317 | 0.3161 | 1.9815 | 0.1523 | 0.3109 |

EFr | 101.5918 | 209.1836 | 0.6067 | 3.4796 | 0.2438 | 0.0172 |

Model | a | b | $\mathit{\mu}$ | $\mathit{\theta}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|

ETIWIW | 1.5665 | 0.0580 | 28.7074 | 16.2632 | - | - |

(0.1228) | (0.0024) | (0.0290) | (0.0292) | - | - | |

TIITFIW | - | 0.1709 | 34.8247 | 5.9024 | - | - |

- | (0.0755) | (2.5283) | (2.2193) | - | - | |

LGFr | 6.6285 | 2.9542 | 21.9999 | 2.0724 | - | - |

(2.0456) | (2.0521) | (1.2151) | (0.1112) | - | - | |

GOFr | - | - | 4.8711 | 12.9533 | 0.0626 | 0.6005 |

- | - | (2.9411) | (1.1160) | (0.2531) | (0.0680) | |

KwFr | 16.8895 | 486.2870 | 5.3100 | 0.3782 | - | - |

(4.0633) | (2.0475) | (2.1463) | (0.0058) | - | - | |

BFr | - | - | 122.7997 | 230.4647 | 88.7582 | 0.1570 |

- | - | (2.1441) | (30.1642) | (7.0361) | (0.1290) | |

EFr | - | - | 11.0292 | 2.6174 | 53.4457 | - |

- | - | (6.0124) | (0.1432) | (5.0211) | - |

**Table 7.**Values of $-\widehat{\ell}$, AIC, W*, A*, K-S and the related p-value for the sum of skin folds data set.

Model | $-\widehat{\mathit{\ell}}$ | AIC | W* | A* | K-S | p-Value |
---|---|---|---|---|---|---|

ETIWIW | 946.3521 | 1900.7040 | 0.0838 | 0.6006 | 0.0645 | 0.3646 |

TIITFIW | 947.5516 | 1901.1030 | 0.0905 | 0.6789 | 0.0663 | 0.3366 |

LGFr | 953.4254 | 1914.8510 | 0.1722 | 1.1660 | 0.0722 | 0.2419 |

GOFr | 959.3028 | 1926.6060 | 0.3388 | 1.9968 | 0.0836 | 0.1181 |

KwFr | 963.5626 | 1935.120 | 0.4373 | 2.5605 | 0.0921 | 0.0645 |

BFr | 955.9227 | 1919.8450 | 0.2539 | 1.5299 | 0.0714 | 0.2542 |

EFr | 955.8263 | 1917.6530 | 0.1998 | 1.4661 | 0.0706 | 0.2660 |

Model | a | b | $\mathit{\mu}$ | $\mathit{\theta}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|

ETIWIW | 2.5869 | 0.1048 | 19.1715 | 21.3741 | - | - |

(0.16813) | (0.4131) | (4.5844) | (3.7627) | - | - | |

TIITFIW | - | 0.4384 | 24.7133 | 7.0868 | - | - |

- | (0.7951) | (6.0332) | (8.6529) | - | - | |

LGFr | 30.2455 | 58.6674 | 26.2278 | 0.8824 | - | - |

(6.9793) | (5.6592) | (1.4252) | (0.1689) | - | - | |

GOFr | - | - | 11.3884 | 26.4729 | 0.1791 | 1.1221 |

- | - | (0.2455) | (2.6733) | (0.0022) | (0.1192) | |

KwFr | 23.4933 | 47.1455 | 9.9122 | 1.5850 | - | - |

(2.2203) | (1.1471) | (2.1669) | (0.2254) | - | - | |

BFr | - | - | 16.4844 | 92.6404 | 72.7783 | 0.6991 |

- | - | (5.9942) | (23.1349) | (8.9369) | (0.0223) | |

EFr | - | - | 18.7691 | 6.0666 | 8.2824 | - |

- | - | (864.5581) | (1.1360) | (4.45652) | - |

**Table 9.**Values of $-\widehat{\ell}$, AIC, W*, A*, K-S and the related p-value for the completed passes of Drew Brees data set.

Model | $-\widehat{\mathit{\ell}}$ | AIC | W* | A* | K-S | p-Value |
---|---|---|---|---|---|---|

ETIWIW | 48.3270 | 104.6542 | 0.0516 | 0.3559 | 0.1219 | 0.9438 |

TIITFIW | 49.6099 | 105.2198 | 0.0532 | 0.3646 | 0.1433 | 0.8974 |

LGFr | 48.4793 | 104.9587 | 0.0523 | 0.3652 | 0.1337 | 0.9369 |

GOFr | 48.3591 | 104.7184 | 0.0545 | 0.3694 | 0.1382 | 0.9199 |

KwFr | 48.3594 | 104.7189 | 0.0559 | 0.3751 | 0.1451 | 0.8891 |

BFr | 48.4446 | 104.8893 | 0.0524 | 0.3639 | 0.1345 | 0.9341 |

EFr | 49.4775 | 104.9550 | 0.0753 | 0.5013 | 0.1686 | 0.7533 |

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Almarashi, A.M.; Elgarhy, M.; Jamal, F.; Chesneau, C.
The Exponentiated Truncated Inverse Weibull-Generated Family of Distributions with Applications. *Symmetry* **2020**, *12*, 650.
https://doi.org/10.3390/sym12040650

**AMA Style**

Almarashi AM, Elgarhy M, Jamal F, Chesneau C.
The Exponentiated Truncated Inverse Weibull-Generated Family of Distributions with Applications. *Symmetry*. 2020; 12(4):650.
https://doi.org/10.3390/sym12040650

**Chicago/Turabian Style**

Almarashi, Abdullah M., Mohammed Elgarhy, Farrukh Jamal, and Christophe Chesneau.
2020. "The Exponentiated Truncated Inverse Weibull-Generated Family of Distributions with Applications" *Symmetry* 12, no. 4: 650.
https://doi.org/10.3390/sym12040650