# Type II Topp–Leone Inverted Kumaraswamy Distribution with Statistical Inference and Applications

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## Abstract

**:**

## 1. Introduction

## 2. TIITLIK Distribution

## 3. Applications

`AdequacyModel`, in which the function

`goodness.fit`was used. We refer to Section 4 for the definitions and theoretical background of the MLEs in the context of the TIITLIK model.

**The first data set (data set 1):**The first data set, given in [19], represents the annual maximum precipitation (in inches) for one rain gauge in Fort Collins (Colorado, USA) from 1900 through 1999. The heading of the data is as follows: 239, 232, 434, 85, 302, 174…

**The second data set (data set 2):**The second data set consists of annual maximum daily precipitation (in unit: mm) at Busan(Korea) from 1904 through 2011 period. The data set has recently been used by [20]. The heading of the data is as follows: 24.8, 140.9, 54.1, 153.5, 47.9, 165.5…

## 4. On the MLEs of the TIITLIK Model

#### 4.1. Definition and Properties

#### 4.2. Simulation

## 5. Mathematical Properties

#### 5.1. Asymptotic Results and Critical Points

#### 5.2. Quantile Function

#### 5.3. Bowley Skewness and Moors Kurtosis

#### 5.4. Mixture Representations

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### 5.5. The Ordinary and Central Moments

#### 5.6. Incomplete Moments

#### 5.7. Weighted Probability Moments

#### 5.8. Stress–Strength Reliability Parameter

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

#### 5.9. Order Statistics

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plots of some probability density functions (pdfs) and hazard rate functions (hrfs) of the type II Topp–Leone (transmuted) inverted Kumaraswamy (TIITLIK) distribution.

**Figure 4.**Plots of the empirical and estimated TIITLIK hrfs and the empirical and estimated TIITLIK chrfs for data set 1.

**Figure 5.**Pots of the empirical and estimated TIITLIK hrfs and the empirical and estimated TIITLIK chrfs for data set 2.

**Figure 8.**Plots for Bowley skewness and Moors kurtosis for $\alpha ,a\in (1,5)$, $b=5$ and $\lambda =-0.3$.

**Figure 9.**Plots for Bowley skewness and Moors kurtosis for $\alpha ,a\in (1,5)$, $b=5$ and $\lambda =-0.8$.

**Figure 10.**Plots for Bowley skewness and Moors kurtosis for $\alpha ,a\in (1,5)$, $b=5$ and $\lambda =0.3$.

**Figure 11.**Plots for Bowley skewness and Moors kurtosis for $\alpha ,a\in (1,5)$, $b=5$ and $\lambda =0.8$.

n | Mean | Median | Standard Deviation | Skewness | Kurtosis | |
---|---|---|---|---|---|---|

Data set 1 | 100 | 175.67 | 158 | 83.17 | 1.32 | 1.71 |

Data set 2 | 105 | 144.6 | 131.6 | 66.18 | 0.93 | 0.73 |

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

TIITLIK | 6.8935 | 0.7683 | 86.5351 | 0.9847 | - | - |

(1.0629) | (0.1939) | (2.2302) | (0.3637) | - | - | |

TLGIK | 1.0146 | - | - | 1.3053 | 89.8279 | 56.2739 |

(3.6755) | - | - | (4.6934) | (4.1124) | (3.8161) | |

MOEIK | 2.5225 | - | - | 566.2283 | 576.4857 | - |

(0.1234) | - | - | (25.8570) | (27.8788) | - | |

GIK | 722.6636 | - | - | - | 0.5796 | 2.3328 |

(9.2161) | - | - | - | (1.3594) | (5.4708) | |

IK | - | 0.7465 | 32.1198 | - | - | - |

- | (0.0409) | (6.0239) | - | - | - |

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

TIITLIK | 35.0662 | 0.6705 | 30.7472 | −0.7928 | - | - |

(3.0642) | (0.1709) | (1.1182) | (0.1795) | - | - | |

TLGIK | 1.8475 | - | - | 0.5927 | 82.5348 | 6.8899 |

(0.5184) | - | - | (0.1581) | (7.0094) | (7.3323) | |

MOEIK | 1.8704 | - | - | 99.9520 | 79.2607 | - |

(0.0934) | - | - | (5.5420) | (6.7107) | - | |

GIK | 351.3337 | - | - | - | 0.6904 | 1.8419 |

(6.0827) | - | - | - | (3.1515) | (8.3987) | |

IK | - | 1.1921 | 241.3032 | - | - | - |

- | (0.0554) | (8.3953) | - | - | - |

Model | $-\widehat{\mathit{\ell}}$ | AIC | BIC | W* | A* | KS | p-Value (KS) |
---|---|---|---|---|---|---|---|

TIITLIK | 565.9951 | 1139.9900 | 1150.4110 | 0.0377 | 0.2893 | 0.0458 | 0.9847 |

TLGIK | 567.5158 | 1143.0320 | 1153.4520 | 0.0644 | 0.4290 | 0.0519 | 0.9502 |

MOEIK | 580.6865 | 1167.3730 | 1175.1890 | 0.0424 | 0.3156 | 0.1544 | 0.0169 |

GIK | 593.7943 | 1193.5890 | 1201.4040 | 0.0472 | 0.3960 | 0.2024 | 0.0005 |

IK | 644.0206 | 1292.0410 | 1297.2520 | 0.0410 | 0.3652 | 0.3018 | 0.00000002 |

Model | $-\widehat{\mathit{\ell}}$ | AIC | BIC | W* | A* | KS | p-Value (KS) |
---|---|---|---|---|---|---|---|

TIITLIK | 582.0684 | 1172.1370 | 1182.7530 | 0.1458 | 0.8300 | 0.0821 | 0.5000 |

TLGIK | 600.0486 | 1208.0970 | 1218.7130 | 0.5001 | 2.8905 | 0.1202 | 0.0958 |

MOEIK | 610.8292 | 1227.6580 | 1235.6200 | 0.1624 | 0.9319 | 0.2033 | 0.0003 |

GIK | 616.0245 | 1238.0490 | 1246.0110 | 0.5327 | 3.0846 | 0.1999 | 0.0004 |

IK | 620.0272 | 1244.0540 | 1249.3620 | 0.4794 | 2.7732 | 0.2147 | 0.0001 |

**Table 6.**MLEs, RMSEs and estimated sf and hrf of the TIITLIK distribution ($a=0.5$, $b=2$, $\alpha =2$ and $\lambda =0.5$).

Parameters | ML | RMSE | $\widehat{\mathit{R}}\left({\mathit{x}}_{0}\right)$ | $\widehat{\mathit{h}}\left({\mathit{x}}_{0}\right)$ | |||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | |||

30 | a | 0.800 | 0.550 | ||||||

b | 2.831 | 1.928 | |||||||

$\alpha $ | 2.518 | 2.037 | 0.736 | 0.515 | 0.371 | 0.301 | 0.264 | 0.221 | |

$\lambda $ | 0.078 | 0.616 | |||||||

50 | a | 0.770 | 0.460 | ||||||

b | 2.610 | 1.586 | |||||||

$\alpha $ | 2.172 | 1.590 | 0.743 | 0.540 | 0.406 | 0.285 | 0.240 | 0.197 | |

$\lambda $ | 0.113 | 0.570 | |||||||

100 | a | 0.763 | 0.358 | ||||||

b | 2.451 | 0.910 | |||||||

$\alpha $ | 1.716 | 1.007 | 0.750 | 0.565 | 0.442 | 0.253 | 0.205 | 0.165 | |

$\lambda $ | 0.184 | 0.469 |

**Table 7.**MLEs, RMSEs, and estimated sf and hrf of the TIITLIK distribution ($a=0.5$, $b=2$, $\alpha =2$ and $\lambda =0.2$).

Parameters | ML | RMSE | $\widehat{\mathit{R}}\left({\mathit{x}}_{0}\right)$ | $\widehat{\mathit{h}}\left({\mathit{x}}_{0}\right)$ | |||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | |||

30 | a | 0.765 | 0.494 | ||||||

b | 3.058 | 2.314 | |||||||

$\alpha $ | 2.436 | 2.519 | 0.816 | 0.622 | 0.478 | 0.262 | 0.240 | 0.204 | |

$\lambda $ | 0.053 | 0.466 | |||||||

50 | a | 0.641 | 0.283 | ||||||

b | 2.374 | 1.232 | |||||||

$\alpha $ | 2.088 | 1.538 | 0.823 | 0.665 | 0.546 | 0.253 | 0.218 | 0.176 | |

$\lambda $ | 0.072 | 0.435 | |||||||

100 | a | 0.652 | 0.265 | ||||||

b | 2.325 | 0.872 | |||||||

$\alpha $ | 1.867 | 1.165 | 0.824 | 0.671 | 0.558 | 0.240 | 0.196 | 0.158 | |

$\lambda $ | 0.071 | 0.390 |

**Table 8.**MLEs, RMSEs and estimated sf and hrf of the TIITLIK distribution ($a=0.5$, $b=2$, $\alpha =3$ and $\lambda =0.5$).

Parameters | ML | RMSE | $\widehat{\mathit{R}}\left({\mathit{x}}_{0}\right)$ | $\widehat{\mathit{h}}\left({\mathit{x}}_{0}\right)$ | |||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | |||

30 | a | 0.879 | 0.644 | ||||||

b | 3.113 | 2.796 | |||||||

$\alpha $ | 3.104 | 2.303 | 0.591 | 0.324 | 0.190 | 0.423 | 0.372 | 0.309 | |

$\lambda $ | 0.266 | 0.930 | |||||||

50 | a | 0.855 | 0.519 | ||||||

b | 2.523 | 1.207 | |||||||

$\alpha $ | 2.584 | 1.668 | 0.615 | 0.382 | 0.253 | 0.338 | 0.285 | 0.235 | |

$\lambda $ | 0.065 | 0.595 | |||||||

100 | a | 0.836 | 0.443 | ||||||

b | 2.507 | 1.081 | |||||||

$\alpha $ | 2.404 | 1.495 | 0.651 | 0.426 | 0.294 | 0.314 | 0.265 | 0.218 | |

$\lambda $ | 0.064 | 0.581 |

**Table 9.**MLEs, RMSEs and estimated sf and hrf of the TIITLIK distribution ($a=0.5$, $b=2$, $\alpha =3$ and $\lambda =0.2$).

Parameters | ML | RMSE | $\widehat{\mathit{R}}\left({\mathit{x}}_{0}\right)$ | $\widehat{\mathit{h}}\left({\mathit{x}}_{0}\right)$ | |||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | |||

30 | a | 0.800 | 0.493 | ||||||

b | 2.808 | 1.775 | |||||||

$\alpha $ | 2.737 | 1.983 | 0.723 | 0.495 | 0.349 | 0.316 | 0.280 | 0.234 | |

$\lambda $ | 0.044 | 0.482 | |||||||

50 | a | 0.767 | 0.446 | ||||||

b | 2.623 | 1.453 | |||||||

$\alpha $ | 2.521 | 1.85 | 0.728 | 0.513 | 0.374 | 0.313 | 0.268 | 0.221 | |

$\lambda $ | 0.068 | 0.466 | |||||||

100 | a | 0.740 | 0.354 | ||||||

b | 2.353 | 0.966 | |||||||

$\alpha $ | 2.320 | 1.496 | 0.738 | 0.541 | 0.410 | 0.283 | 0.236 | 0.194 | |

$\lambda $ | −0.00542 | 0.449 |

**Table 10.**MLEs, RMSEs and estimated sf and hrf of the TIITLIK distribution ($a=1.2$, $b=3$, $\alpha =2$ and $\lambda =0.5$).

Parameters | ML | RMSE | $\widehat{\mathit{R}}\left({\mathit{x}}_{0}\right)$ | $\widehat{\mathit{h}}\left({\mathit{x}}_{0}\right)$ | |||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{3}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{5}$ | ${\mathit{x}}_{\mathbf{0}}=\mathbf{7}$ | |||

30 | a | 1.530 | 0.898 | ||||||

b | 4.257 | 2.831 | |||||||

$\alpha $ | 2.261 | 1.682 | 0.151 | 0.032 | 0.010 | 0.269 | 0.278 | 0.252 | |

$\lambda $ | 0.705 | 0.958 | |||||||

50 | a | 1.504 | 0.778 | ||||||

b | 3.945 | 2.433 | |||||||

$\alpha $ | 2.068 | 1.486 | 0.191 | 0.052 | 0.020 | 0.265 | 0.257 | 0.229 | |

$\lambda $ | 0.620 | 0.917 | |||||||

100 | a | 1.464 | 0.721 | ||||||

b | 3.633 | 1.794 | |||||||

$\alpha $ | 2.057 | 1.204 | 0.229 | 0.073 | 0.031 | 0.262 | 0.252 | 0.223 | |

$\lambda $ | 0.481 | 0.439 |

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## Share and Cite

**MDPI and ACS Style**

ZeinEldin, R.A.; Jamal, F.; Chesneau, C.; Elgarhy, M.
Type II Topp–Leone Inverted Kumaraswamy Distribution with Statistical Inference and Applications. *Symmetry* **2019**, *11*, 1459.
https://doi.org/10.3390/sym11121459

**AMA Style**

ZeinEldin RA, Jamal F, Chesneau C, Elgarhy M.
Type II Topp–Leone Inverted Kumaraswamy Distribution with Statistical Inference and Applications. *Symmetry*. 2019; 11(12):1459.
https://doi.org/10.3390/sym11121459

**Chicago/Turabian Style**

ZeinEldin, Ramadan A., Farrukh Jamal, Christophe Chesneau, and Mohammed Elgarhy.
2019. "Type II Topp–Leone Inverted Kumaraswamy Distribution with Statistical Inference and Applications" *Symmetry* 11, no. 12: 1459.
https://doi.org/10.3390/sym11121459