# Beta Exponentiated Modified Weibull Distribution: Properties and Application

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## Abstract

**:**

## 1. Introduction

## 2. Beta Exponentiated Modified Weibull Distribution

#### 2.1. Special Cases of the BEMW Distribution

#### 2.1.1. Beta Exponentiated Weibull (BEW) Distribution

#### 2.1.2. Beta Generalized Rayleigh (BGR) Distribution

#### 2.1.3. Beta Modified Weibull (BMW) Distribution

#### 2.1.4. Beta Rayleigh (BR) Distribution

#### 2.1.5. Beta Exponentiated Exponential (BEE) Distribution

#### 2.1.6. Beta Generalized Linear Failure Rate (BGLFR) Distribution

#### 2.1.7. Exponentiated Modified Weibull (EMW) Distribution

## 3. Expansion of the Probability Distribution and Density Function

**Proof.**

## 4. Statistical Properties

#### Moments

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 5. Order Statistics of the Beta Exponentiated Modified Weibull Distribution

## 6. Reliability Analysis

#### 6.1. Survival Function

#### 6.2. Hazard Function

## 7. Parameter Estimation

#### 7.1. Maximum Likelihood Estimation

#### 7.2. Least Squares Estimation

## 8. Application

## 9. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Various shapes of the pdfs of the beta exponentiated modified Weibull (BEMW) distribution for the varying values of the parameters.

**Figure 2.**Special cases of the beta exponentiated modified Weibull (BEMW) distribution (In figure, D stands for the ‘distribution’).

Length | Average | Minimum | Maximum | ${\mathit{Q}}_{1}$ | Median | ${\mathit{Q}}_{3}$ | Standard Deviation |
---|---|---|---|---|---|---|---|

40 | 6.2525 | 1.6 | 9.0 | 5.075 | 6.50 | 7.825 | 1.95553 |

**Table 2.**Maximum likelihood estimates with standard errors and distribution selection measures for the time-to-failure of the turbocharger data.

Model | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\delta}}$ | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\theta}}$ | AIC | BIC | AICc |
---|---|---|---|---|---|---|---|---|---|

BEMWD | 2.6629 | 0.9664 | 2.9256 | 0.2769 | 1.6231 | 0.3785 | 24.99 | 35.13 | 27.54 |

(0.5594) | (2.0912) | (0.5068) | (7.7832) | (37.7734) | (23.1998) | ||||

EMWD | - | - | 26.9086 | 7.6903 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-7}$ | 15.1406 | 0.01399 | 28.69 | 35.44 | 29.83 |

(0.0916) | (15.0308) | (0.5086) | (742.7817) | ||||||

MWD | - | - | - | 0.1895 | 0.2810 | 4.0529 $\times \phantom{\rule{0.166667em}{0ex}}{10}^{-6}$ | 34.34 | 39.41 | 35.01 |

(7.3718) | (3.3962) | (2775.03) | |||||||

ED | - | - | - | - | - | 0.1817 | 30.56 | 32.25 | 30.66 |

(8.0248) |

**Table 3.**Least squares (LS) estimates with the standard errors and distribution selection measures for the time-to-failure of the turbocharger data.

Model | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\gamma}}$ | $\widehat{\mathit{\delta}}$ | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\theta}}$ | AIC | BIC | AICc |
---|---|---|---|---|---|---|---|---|---|

BEMWD | 1.4268 | 1.3079 | 1.500 | 0.9999 | 1.9999 | 1.8999 | 18.34 | 28.47 | 20.89 |

(3.7841) | (0.3475) | (0.0003) | (0.0019) | (0.0119) | (0.0115) | ||||

EMWD | - | - | 1.5129 | 0.9689 | 1.8378 | 1.7407 | 34.34 | 41.10 | 35.48 |

(0.0006) | (0.0483) | (0.1911) | (0.2035) | ||||||

MWD | - | - | - | 0.9893 | 1.9366 | 1.8392 | 32.34 | 37.41 | 33.01 |

(0.0179) | (0.0920) | (0.0925) | |||||||

ED | - | - | - | - | - | 0.1353 | 24.56 | 26.25 | 24.66 |

(9.8036) |

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**MDPI and ACS Style**

Shahzad, M.N.; Ullah, E.; Hussanan, A.
Beta Exponentiated Modified Weibull Distribution: Properties and Application. *Symmetry* **2019**, *11*, 781.
https://doi.org/10.3390/sym11060781

**AMA Style**

Shahzad MN, Ullah E, Hussanan A.
Beta Exponentiated Modified Weibull Distribution: Properties and Application. *Symmetry*. 2019; 11(6):781.
https://doi.org/10.3390/sym11060781

**Chicago/Turabian Style**

Shahzad, Mirza Naveed, Ehsan Ullah, and Abid Hussanan.
2019. "Beta Exponentiated Modified Weibull Distribution: Properties and Application" *Symmetry* 11, no. 6: 781.
https://doi.org/10.3390/sym11060781