# Type II Power Topp-Leone Generated Family of Distributions with Statistical Inference and Applications

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

- create distributions with different shapes for the pdf and hrf,
- transform symmetrical distributions into skewed distributions,
- construct heavy-tailed distributions,
- increase the flexibility of the mode(s), mean, variance, skewness and kurtosis of the baseline distribution,
- provide better fits than other general families (including those based on the Topp-Leone distribution) with the same baseline distribution and possibly more complex (with more parameters).

## 2. Definition of the TIIPTL-G Family

#### 2.1. Important Functions

#### 2.2. Asymptotes and Shapes

#### 2.3. Special Members of the Family

#### 2.4. The TIIPTLIEx Distribution

## 3. Some Properties of the TIIPTL-G Family

#### 3.1. Quantile Function

#### 3.2. Mixture Representation

**Theorem**

**1.**

- we have the following expansion for$F(x;\alpha ,\beta ,\xi )$:$$F(x;\alpha ,\beta ,\xi )=1+\sum _{k,\ell =0}^{+\infty}{a}_{k,\ell}{G}_{\ell}(x;\xi )$$$${a}_{k,\ell}=\left(\genfrac{}{}{0pt}{}{\alpha}{k}\right)\left(\genfrac{}{}{0pt}{}{\beta (k+\alpha )}{\ell}\right){(-1)}^{k+\ell +1}{2}^{\alpha -k}.$$
- we have the following expansion for$f(x;\alpha ,\beta ,\xi )$:$$f(x;\alpha ,\beta ,\xi )=\sum _{k=0}^{+\infty}\sum _{\ell =1}^{+\infty}{a}_{k,\ell}{g}_{\ell}(x;\xi ).$$

**Proof.**

#### 3.3. Some Kinds of Moments

#### 3.3.1. Moments and Central Moments

#### 3.3.2. Inverted Moments

#### 3.3.3. Incomplete Moments

#### 3.4. Stochastic Ordering

**Proposition**

**1.**

**Proof.**

#### 3.5. Reliability

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### 3.6. Order Statistics

**Proposition**

**4.**

**Proof.**

## 4. Estimation and Simulation

#### 4.1. Maximum Likelihood Method of Estimation

#### 4.2. Percentile Method of Estimation

#### 4.3. Right-Tail Anderson-Darling Method of Estimation

#### 4.4. A Simulation Study

## 5. Applications

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Topp, C.W.; Leone, F.C. A Family of J-Shaped Frequency Functions. J. Am. Stat. Assoc.
**1955**, 50, 209–219. [Google Scholar] [CrossRef] - Ghitany, M.E.; Kotz, S.; Xie, M. On some reliability measures and their stochastic orderings for the Topp–Leone distribution. J. Appl. Stat.
**2005**, 32, 715–722. [Google Scholar] [CrossRef] - MirMostafaee, S. On the moments of order statistics coming from the Topp–Leone distribution. Stat. Probab. Lett.
**2014**, 95, 85–91. [Google Scholar] [CrossRef] - Nadarajah, S.; Kotz, S. Moments of some J-shaped distributions. J. Appl. Stat.
**2003**, 30, 311–317. [Google Scholar] [CrossRef] - Bayoud, H.A. Estimating the shape parameter of Topp-Leone distribution based on progressive type II censored samples. REVSTAT-Stat. J.
**2016**, 14, 415–431. [Google Scholar] - Feroze, N.; Aslam, M. On Bayesian analysis of failure rate under Topp Leone distribution using complete and censored samples. Int. J. Math. Comput. Phys. Electr. Comput. Eng.
**2013**, 7, 426–432. [Google Scholar] - Sehgal, S.; Pushkarna, N.; Saran, J. Exact moments of order statistics, MLE, L-moments and TL-moments estimation from Topp-Leone distribution. Int. J. Math. Stat.
**2019**, 20, 37–50. [Google Scholar] - Sultan, H.; Ahmad, S.P. Bayesian Analysis of Topp-Leone Distribution under Different Loss Functions and Different Priors. J. Stat. Appl. Probab. Lett.
**2016**, 3, 109–118. [Google Scholar] [CrossRef] - Pourdarvish, A.; Mirmostafaee, S.M.T.K.; Naderi, K. The exponentiated Topp-Leone distribution: Properties and application. J. Appl. Environ. Biol. Sci.
**2015**, 5, 251–256. [Google Scholar] - ZeinEldin, R.A.; Chesneau, C.; Jamal, F.; Elgarhy, M. Different Estimation Methods for Type I Half-Logistic Topp–Leone Distribution. Mathematics
**2019**, 7, 985. [Google Scholar] [CrossRef] [Green Version] - Al-Shomrani, A.; Arif, O.; Hanif, S.; Shahbaz, M.Q.; Shawky, A. Topp–Leone Family of Distributions: Some Properties and Application. Pak. J. Stat. Oper. Res.
**2016**, 12, 443. [Google Scholar] [CrossRef] [Green Version] - Jamal, Q.A.; Arshad, M. Statistical Inference for Topp–Leone-generated Family of Distributions Based on Records. J. Stat. Theory Appl.
**2019**, 18, 65–78. [Google Scholar] [CrossRef] - Mahdavi, A. Generalized Topp-Leone family of distributions. J. Biostat. Epidemiol.
**2017**, 3, 65–75. [Google Scholar] - Sangsanit, Y.; Bodhisuwan, W. The Topp-Leone generator of distributions: Properties and inferences. Songklanakarin J. Sci. Technol.
**2016**, 38, 537–548. [Google Scholar] - Kunjiratanachot, N.; Bodhisuwan, W.; Volodin, A. The Topp-Leone generalized exponential power series distribution with applications. J. Probab. Stat. Sci.
**2018**, 16, 197–208. [Google Scholar] - Roozegar, R.; Nadarajah, S. A New Class of Topp–Leone Power Series Distributions with Reliability Application. J. Fail. Anal. Prev.
**2017**, 17, 955–970. [Google Scholar] [CrossRef] - Elgarhy, M.; Nasir, M.A.; Jamal, F.; Ozel, G. The type II Topp-Leone generated family of distributions: Properties and applications. J. Stat. Manag. Syst.
**2018**, 21, 1529–1551. [Google Scholar] [CrossRef] - Brito, E.; Cordeiro, G.M.; Yousof, H.M.; Alizadeh, M.; Silva, G.O. The Topp–Leone odd log-logistic family of distributions. J. Stat. Comput. Simul.
**2017**, 87, 1–19. [Google Scholar] [CrossRef] - Hassan, A.S.; Elgarhy, M.; Ahmad, Z. Type II generalized Topp-Leone family of distributions: Properties and applications. J. Data Sci.
**2019**, 17, 638–659. [Google Scholar] - Reyad, H.; Korkmaz, M.Ç.; Afify, A.Z.; Hamedani, G.G.; Othman, S. The Fréchet Topp Leone-G Family of Distributions: Properties, Characterizations and Applications. Ann. Data Sci.
**2019**, 1–22. [Google Scholar] [CrossRef] - Reyad, H.M.; Alizadeh, M.; Jamal, F.; Othman, S.; Hamedani, G.G. The Exponentiated Generalized Topp Leone-G Family of Distributions: Properties and Applications. Pak. J. Stat. Oper. Res.
**2019**, 15, 1–24. [Google Scholar] [CrossRef] [Green Version] - Yousof, H.M.; Alizadeh, M.; Jahanshahi, S.M.A.; Ramires, T.G.; Ghosh, I.; Hamedani, G.G. The transmuted Topp-Leone G family of distributions: Theory, characterizations and applications. J. Data Sci.
**2017**, 15, 723–740. [Google Scholar] - Rezaei, S.; Sadr, B.B.; Alizadeh, M.; Nadarajah, S. Topp-Leone generated family of distributions: Properties and applications. Commun. Stat. Theory Methods
**2016**, 46, 2893–2909. [Google Scholar] [CrossRef] - Keller, A.; Kamath, A.; Perera, U. Reliability analysis of CNC machine tools. Reliab. Eng.
**1982**, 3, 449–473. [Google Scholar] [CrossRef] - Lin, C.; Duran, B.; Lewis, T. Inverted gamma as a life distribution. Microelectron. Reliab.
**1989**, 29, 619–626. [Google Scholar] [CrossRef] - Shaked, M.; Shanthikumar, J.G. Stochastic Orders and their Applications; Academic Press: London, UK, 1994. [Google Scholar]
- Pugh, E.L. The Best Estimate of Reliability in the Exponential Case. Oper. Res.
**1963**, 11, 57–61. [Google Scholar] [CrossRef] - R Development Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Austria, Vienna, 2009. [Google Scholar]
- Kao, J.H.K. Computer Methods for Estimating Weibull Parameters in Reliability Studies. IRE Trans. Reliab. Qual. Control.
**1958**, 13, 15–22. [Google Scholar] [CrossRef] - Anderson, T.W.; Darling, D.A. Asymptotic Theory of Certain “Goodness of Fit” Criteria Based on Stochastic Processes. Ann. Math. Stat.
**1952**, 23, 193–212. [Google Scholar] [CrossRef] - Oguntunde, P.E.; Babatunde, O.S.; Ogunmola, A.O. Theoretical analysis of the Kumaraswamy-inverse exponential distribution. Int. J. Stat. Appl.
**2014**, 4, 113–116. [Google Scholar] - Khan, M.S. The beta inverseWeibull distribution. Int. Trans. Math. Sci. Comput.
**2010**, 3, 113–119. [Google Scholar] - Ramadan, D.; Magdy, W. On the Alpha-Power Inverse Weibull Distribution. Int. J. Comput. Appl.
**2018**, 181, 6–12. [Google Scholar] - Oguntunde, P.E.; Adejumo, A.O.; Khaleel, M.A.; Okagbue, H.I.; Odetunmibi, O.A. The logistic inverse exponential distribution: Basic structural properties and application. In Proceedings of the World Congress on Engineering and Computer Science 2018 (Vol II WCECS 2018), San Francisco, CA, USA, 23–25 October 2018. [Google Scholar]
- Aldahlan, M.A. The inverse Weibull inverse exponential distribution with application. Int. J. Contemp. Math. Sci.
**2019**, 14, 17–30. [Google Scholar] [CrossRef] [Green Version] - Yahia, N.; Mohammed, H.F. The type II Topp Leone generalized inverse Rayleigh distribution. Int. J. Contemp. Math. Sci.
**2019**, 14, 113–122. [Google Scholar] [CrossRef] - Crowder, M.J.; Kimber, A.C.; Smith, R.L.; Sweeting, T.J. Statistical Analysis of Reliability Data; Chapman and Hall: London, UK, 1991. [Google Scholar]
- Aryal, G.R.; Edwin, M.O.; Hamedani, G.G.; Haitham, M.Y. The Topp-Leone generatedWeibull distribution: Regression model, characterizations and applications. Int. J. Stat. Probab.
**2017**, 6, 126–141. [Google Scholar] [CrossRef] - Pešta, M. Total least squares and bootstrapping with application in calibration. Stat. A J. Theor. Appl. Stat.
**2013**, 47, 966–991. [Google Scholar] [CrossRef] - Peštová, B.; Pešta, M. Change Point Estimation in Panel Data without Boundary Issue. Risks
**2017**, 5, 7. [Google Scholar] [CrossRef]

Distribution | Name of $\mathit{G}(\mathit{x};\mathit{\xi})$ | $\mathit{\xi}$ | Support | $\mathit{S}(\mathit{x};\mathit{\alpha},\mathit{\beta},\mathit{\xi})$ |
---|---|---|---|---|

TIIPTLU | Uniform | $\left(\theta \right)$ | $(0,\theta )$ | ${\left[1-\frac{x}{\theta}\right]}^{\alpha \beta}{\left\{2-{\left[1-\frac{x}{\theta}\right]}^{\beta}\right\}}^{\alpha}$ |

TIIPTLP | Power | $\left(a\right)$ | $(0,1)$ | ${\left[1-{x}^{a}\right]}^{\alpha \beta}{\left\{2-{\left[1-{x}^{a}\right]}^{\beta}\right\}}^{\alpha}$ |

TIIPTLIEx | Inverse exponential | $\left(\theta \right)$ | $(0,+\infty )$ | ${\left[1-{e}^{-\theta /x}\right]}^{\alpha \beta}{\left\{2-{\left[1-{e}^{-\theta /x}\right]}^{\beta}\right\}}^{\alpha}$ |

TIIPTLD | Dagum | $(a,b,c)$ | $(0,+\infty )$ | ${\left[1-{\left(1+{\left(\frac{x}{a}\right)}^{-b}\right)}^{-c}\right]}^{\alpha \beta}{\left\{2-{\left[1-{\left(1+{\left(\frac{x}{a}\right)}^{-b}\right)}^{-c}\right]}^{\beta}\right\}}^{\alpha}$ |

TIIPTLFr | Fréchet | $\left(a\right)$ | $(0,+\infty )$ | ${\left[1-{e}^{-{x}^{-a}}\right]}^{\alpha \beta}{\left\{2-{\left[1-{e}^{-{x}^{-a}}\right]}^{\beta}\right\}}^{\alpha}$ |

TIIPTLHC | Half Cauchy | $\left(a\right)$ | $(0,+\infty )$ | ${\left[1-\frac{2}{\pi}\mathrm{arctan}\left(\frac{x}{a}\right)\right]}^{\alpha \beta}{\left\{2-{\left[1-\frac{2}{\pi}\mathrm{arctan}\left(\frac{x}{a}\right)\right]}^{\beta}\right\}}^{\alpha}$ |

TIIPTLo | Logistic | $(a,b)$ | $\mathbb{R}$ | ${\left[1-\frac{1}{1+{e}^{-(x-a)/b}}\right]}^{\alpha \beta}{\left\{2-{\left[1-\frac{1}{1+{e}^{-(x-a)/b}}\right]}^{\beta}\right\}}^{\alpha}$ |

TIIPTLN | Normal | $(\mu ,\sigma )$ | $\mathbb{R}$ | ${\left[1-\Phi (x;\mu ,\sigma )\right]}^{\alpha \beta}{\left\{2-{\left[1-\Phi (x;\mu ,\sigma )\right]}^{\beta}\right\}}^{\alpha}$ |

**Table 2.**Estimates and MSEs of TIIPTLIEx distribution for MLE, PCand RTADestimates for Set1: $(\alpha =2,\theta =1.5,\beta =2)$.

MLEs | PCEs | RTADEs | ||||
---|---|---|---|---|---|---|

$\mathit{n}$ | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs |

50 | 2.143 | 0.291 | 2.094 | 0.662 | 1.599 | 0.801 |

1.528 | 0.067 | 1.392 | 0.102 | 1.115 | 0.585 | |

2.022 | 0.271 | 1.857 | 0.370 | 1.572 | 0.707 | |

100 | 2.115 | 0.106 | 2.176 | 0.480 | 1.668 | 0.786 |

1.519 | 0.033 | 1.405 | 0.072 | 1.142 | 0.570 | |

1.960 | 0.066 | 1.826 | 0.286 | 1.567 | 0.673 | |

200 | 2.088 | 0.074 | 2.003 | 0.393 | 1.626 | 0.759 |

1.511 | 0.011 | 1.452 | 0.033 | 1.143 | 0.550 | |

1.981 | 0.042 | 1.968 | 0.142 | 1.585 | 0.653 | |

500 | 2.105 | 0.047 | 2.089 | 0.221 | 1.813 | 0.488 |

1.499 | 4.844 * | 1.428 | 0.017 | 1.282 | 0.344 | |

1.948 | 0.025 | 1.846 | 0.133 | 1.741 | 0.406 |

**Table 3.**Estimates and MSEs of TIIPTLIEx distribution for MLE, PC and RTAD estimates for Set2: $(\alpha =3,\theta =1.5,\beta =2)$.

MLEs | PCEs | RTADEs | ||||
---|---|---|---|---|---|---|

$\mathit{n}$ | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs |

50 | 2.489 | 0.365 | 2.448 | 1.084 | 1.863 | 2.770 |

1.573 | 0.050 | 1.315 | 0.177 | 1.078 | 0.814 | |

2.441 | 0.324 | 1.984 | 0.611 | 1.795 | 1.210 | |

100 | 2.505 | 0.315 | 2.346 | 1.008 | 1.803 | 2.512 |

1.572 | 0.034 | 1.395 | 0.062 | 1.083 | 0.687 | |

2.451 | 0.270 | 2.177 | 0.330 | 1.765 | 0.944 | |

200 | 2.466 | 0.307 | 2.339 | 0.804 | 1.920 | 2.136 |

1.539 | 0.013 | 1.468 | 0.025 | 1.147 | 0.573 | |

2.364 | 0.161 | 2.317 | 0.289 | 1.835 | 0.789 | |

500 | 2.482 | 0.279 | 2.218 | 0.638 | 1.919 | 2.127 |

1.559 | 7.459 * | 1.470 | 0.016 | 1.165 | 0.572 | |

2.384 | 0.156 | 2.325 | 0.188 | 1.874 | 0.799 |

**Table 4.**Estimates and MSEs of TIIPTLIEx distribution for MLE, PC and RTAD estimates for Set3: $(\alpha =2,\theta =1.5,\beta =3)$.

MLEs | PCEs | RTADEs | ||||
---|---|---|---|---|---|---|

$\mathit{n}$ | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs |

50 | 2.630 | 0.525 | 2.439 | 0.463 | 2.661 | 0.620 |

1.536 | 0.040 | 1.389 | 0.121 | 1.489 | 0.060 | |

2.589 | 0.318 | 2.453 | 0.821 | 2.481 | 0.469 | |

100 | 2.585 | 0.432 | 2.334 | 0.250 | 2.626 | 0.469 |

1.517 | 0.019 | 1.395 | 0.074 | 1.452 | 0.022 | |

2.627 | 0.312 | 2.465 | 0.704 | 2.429 | 0.386 | |

200 | 2.534 | 0.310 | 2.344 | 0.239 | 2.593 | 0.389 |

1.474 | 0.012 | 1.363 | 0.050 | 1.429 | 0.013 | |

2.507 | 0.285 | 2.319 | 0.694 | 2.368 | 0.383 | |

500 | 2.524 | 0.292 | 2.344 | 0.164 | 2.619 | 0.376 |

1.454 | 6.626 * | 1.406 | 0.027 | 1.449 | 7.531* | |

2.481 | 0.283 | 2.432 | 0.444 | 2.392 | 0.382 |

**Table 5.**Estimates and MSEs of TIIPTLIEx distribution for MLE, PC and RTAD estimates for Set4: $(\alpha =3,\theta =1.5,\beta =3)$.

MLEs | PCEs | RTADEs | ||||
---|---|---|---|---|---|---|

$\mathit{n}$ | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs |

50 | 3.224 | 0.652 | 2.711 | 0.464 | 3.120 | 0.318 |

1.567 | 0.058 | 1.421 | 0.099 | 1.514 | 0.047 | |

3.193 | 0.494 | 3.022 | 0.911 | 3.082 | 0.266 | |

100 | 3.336 | 0.431 | 2.582 | 0.311 | 3.082 | 0.221 |

1.542 | 0.016 | 1.369 | 0.053 | 1.510 | 0.027 | |

3.027 | 0.187 | 2.838 | 0.436 | 3.022 | 0.148 | |

200 | 3.280 | 0.309 | 2.705 | 0.159 | 3.019 | 0.099 |

1.522 | 7.876 * | 1.418 | 0.027 | 1.511 | 0.013 | |

2.965 | 0.118 | 2.912 | 0.248 | 3.030 | 0.073 | |

500 | 3.225 | 0.196 | 2.760 | 0.091 | 3.091 | 0.082 |

1.480 | 3.803 * | 1.476 | 8.344 * | 1.513 | 5.006 * | |

2.817 | 0.117 | 3.065 | 0.099 | 2.995 | 0.050 |

Sets | n | MLEs | PCEs | RTADEs |
---|---|---|---|---|

Set1: $(\alpha =2,\theta =1.5,\beta =2)$ | 50 | 1.0 | 2.0 | 3.0 |

100 | 1.0 | 2.0 | 3.0 | |

200 | 1.0 | 2.0 | 3.0 | |

500 | 1.0 | 2.0 | 3.0 | |

Set2: $(\alpha =3,\theta =1.5,\beta =2)$ | 50 | 1.0 | 2.5 | 2.5 |

100 | 1.0 | 2.0 | 3.0 | |

200 | 1.0 | 2.0 | 3.0 | |

500 | 1.0 | 2.0 | 3.0 | |

Set3: $(\alpha =2,\theta =1.5,\beta =3)$ | 50 | 1.0 | 2.5 | 2.5 |

100 | 1.0 | 2.5 | 2.5 | |

200 | 1.0 | 2.5 | 2.5 | |

500 | 1.0 | 2.5 | 2.5 | |

Set4: $(\alpha =3,\theta =1.5,\beta =3)$ | 50 | 2.0 | 3.0 | 1.0 |

100 | 2.0 | 3.0 | 1.0 | |

200 | 2.0 | 3.0 | 1.0 | |

500 | 2.5 | 2.5 | 1.0 | |

Sum of the partial ranks | 20.5 | 38.0 | 37.5 | |

Final rank | 1.0 | 3.0 | 2.0 |

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

TIIPTLIEx | 46.4917 | 0.2570 | 58.4781 | - | - | - |

(2.0811) | (0.4493) | (3.7038) | - | - | - | |

KIEx | 15.5780 | 5.6385 | 8.6130 | - | - | - |

(2.7977) | (2.2527) | (0.5530) | - | - | - | |

BIEx | - | - | 11.6360 | - | 16.9072 | 3.8280 |

- | - | (0.9004) | - | (0.1398) | (1.0557) | |

AIW | 35.4934 | 1.2320 | - | 33.8531 | - | - |

(3.0404) | (0.1345) | - | (6.8240) | - | ||

LIEx | - | - | - | - | 4.2022 | 28.3120 |

- | - | - | - | (0.7476) | (2.9606) | |

IWIEx | 1.7362 | 1.8178 | 27.5001 | - | - | - |

(2.4572) | (0.2659) | (9.7337) | - | - | - | |

TIR | 14.5765 | 5.6554 | 1.0716 | - | - | - |

(0.6273) | (1.38163) | (0.3041) | - | - | - | |

IEx | - | - | 55.4934 | - | - | - |

- | - | (11.3275) | - | - | - |

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

TIIPTLIEx | 28.7430 | 0.1522 | 10.9219 | - | - | - |

(0.4839) | (0.1831) | (2.4038) | - | - | - | |

KIEx | 21.3327 | 1.7608 | 1.3437 | - | - | - |

(1.9516) | (0.3244) | (7.9917) | - | - | - | |

BIEx | - | - | 11.7217 | - | 2.8252 | 1.6669 |

- | - | (0.1304) | - | (0.2676) | (0.9539) | |

AIW | 3.6810 | 49.2020 | - | 1.4616 | - | - |

(4.6113) | (0.0244) | - | (0.1661) | - | ||

LIEx | - | - | - | - | 2.4594 | 11.2381 |

- | - | - | - | (0.2792) | (1.2269) | |

IWIEx | 1.2470 | 1.2812 | 14.3304 | - | - | - |

(2.1153) | (0.1442) | (5.3741) | - | - | - | |

TIR | 16.8288 | 0.2966 | 0.5422 | - | - | - |

(9.7095) | (3.8675) | (0.0809) | - | - | - | |

IEx | - | - | 20.4026 | - | - | - |

- | - | (2.5503) | - | - | - |

Model | $-\widehat{\mathit{\ell}}$ | AIC | CAIC | BIC | HQIC | W* | A* |
---|---|---|---|---|---|---|---|

TIIPTLIEx | 117.7402 | 241.4803 | 242.6803 | 245.0145 | 242.4180 | 0.0343 | 0.2247 |

KIEx | 118.0351 | 242.0702 | 243.2702 | 245.6044 | 243.0078 | 0.0373 | 0.2561 |

BIEx | 118.7151 | 243.4301 | 244.6301 | 246.9643 | 244.3677 | 0.0499 | 0.3567 |

AIW | 125.6581 | 257.3162 | 258.5162 | 260.8504 | 258.2539 | 0.0399 | 0.2780 |

LIEx | 119.1395 | 242.2790 | 242.8504 | 245.6351 | 242.9043 | 0.0526 | 0.3747 |

IWIEx | 120.6731 | 247.3462 | 248.5462 | 250.8804 | 248.2838 | 0.0910 | 0.6401 |

TIR | 120.6591 | 247.3182 | 248.5182 | 250.8524 | 248.2558 | 0.0947 | 0.6639 |

IEx | 126.9634 | 255.9268 | 256.1086 | 257.1049 | 256.2394 | 0.0498 | 0.3562 |

Model | $-\widehat{\mathit{\ell}}$ | AIC | CAIC | BIC | HQIC | W* | A* |
---|---|---|---|---|---|---|---|

TIIPTLIEx | 293.7982 | 593.5963 | 593.9963 | 600.0730 | 596.1478 | 0.1211 | 0.8804 |

KIEx | 294.761 | 595.5237 | 595.9237 | 602.0003 | 598.0751 | 0.1512 | 1.0674 |

BIEx | 294.8518 | 595.7036 | 596.1036 | 602.1802 | 598.2557 | 0.1575 | 1.1037 |

AIW | 294.9399 | 595.8798 | 596.2798 | 602.3565 | 598.4313 | 0.1591 | 1.1133 |

LIEx | 299.3795 | 602.7589 | 602.9556 | 607.0767 | 604.4599 | 0.2466 | 1.6523 |

IWIEx | 295.5628 | 597.1256 | 597.5256 | 603.6023 | 599.6771 | 0.1889 | 1.28434 |

TIR | 296.7181 | 599.4362 | 599.8362 | 605.9128 | 601.9876 | 0.2378 | 1.5590 |

IEx | 299.1754 | 600.3507 | 600.4152 | 602.5096 | 601.2012 | 0.1566 | 1.0983 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bantan, R.A.R.; Jamal, F.; Chesneau, C.; Elgarhy, M.
Type II Power Topp-Leone Generated Family of Distributions with Statistical Inference and Applications. *Symmetry* **2020**, *12*, 75.
https://doi.org/10.3390/sym12010075

**AMA Style**

Bantan RAR, Jamal F, Chesneau C, Elgarhy M.
Type II Power Topp-Leone Generated Family of Distributions with Statistical Inference and Applications. *Symmetry*. 2020; 12(1):75.
https://doi.org/10.3390/sym12010075

**Chicago/Turabian Style**

Bantan, Rashad A. R., Farrukh Jamal, Christophe Chesneau, and Mohammed Elgarhy.
2020. "Type II Power Topp-Leone Generated Family of Distributions with Statistical Inference and Applications" *Symmetry* 12, no. 1: 75.
https://doi.org/10.3390/sym12010075