# A Bivariate Extension to Exponentiated Inverse Flexible Weibull Distribution: Shock Model, Features, and Inference to Model Asymmetric Data

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## Abstract

**:**

## 1. Introduction

## 2. Structure of the BEIFWE Model

## 3. Distributional Properties

#### 3.1. Joint Reliability and Joint (Reversed) Hazard Rate Functions

#### 3.2. Marginal Probability Density Functions

**Lemma**

**1.**

**Proof.**

#### 3.3. The Distribution of $Y=max\{{X}_{1},{X}_{2}\}$ and $W=min\{{X}_{1},{X}_{2}\}$

#### 3.4. Conditional Probability Density Functions

**Lemma**

**2.**

**Proof.**

#### 3.5. Marginal Expectation

**Lemma**

**3.**

**Proof.**

## 4. Maximum Likelihood Estimation (MLE)

## 5. MLE Performance: A Simulation Study

**R**software package. The primary aim of this section is to introduce an assessment of the properties of the MLE in terms of bias and mean-squared error (MSE) for the parameters. To test the performance of the MLE technique, two schemes are considered and discussed under different sample sizes. The experimental schemas can be formulated as follows:

- Schema I: BEIFWE($0.5,0.7,0.9,0.2,0.5$);
- Schema II: BEIFWE($0.2,0.4,0.6,0.8,1.1$).

## 6. Comparative Study: Statistics and Real Data Analysis

#### 6.1. Dataset I: Football Data

#### 6.2. Dataset II: Motor Data

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**(

**Top**) simulation results of the BEIFWE (0.5,0.7,0.9,0.2,0.5) model; (

**bottom**) simulation results of the BEIFWE (0.2,0.4,0.6,0.8,1.1) model.

Model | $\widehat{\mathit{a}}$ | $\widehat{\mathit{b}}$ | $\widehat{{\mathit{\lambda}}_{1}}$ | $\widehat{{\mathit{\lambda}}_{2}}$ | $\widehat{{\mathit{\lambda}}_{3}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ |
---|---|---|---|---|---|---|---|

BW | $0.0837$ | − | $0.3974$ | $0.2738$ | $0.3389$ | − | − |

BGPW | $0.0377$ | − | $3.2294$ | $1.9831$ | $4.0840$ | − | − |

BGz | $0.0406$ | − | $0.0036$ | $0.0023$ | $0.0213$ | − | − |

BBUXGz | $0.0063$ | $0.0154$ | $0.1320$ | $0.1873$ | $0.2014$ | − | − |

BGE | $0.0393$ | − | $1.5532$ | $0.4993$ | $1.1563$ | − | − |

MOBE | − | − | $0.0121$ | $0.0141$ | $0.0221$ | − | − |

BEW | $0.0123$ | $1.2683$ | $1.2269$ | $0.3820$ | $0.6611$ | − | − |

BGuE | $5.0111$ | $4.0814$ | $2.6784$ | $0.9621$ | $2.0653$ | − | − |

BGLFR | $0.0002$ | $0.0008$ | $0.4520$ | $0.1567$ | $0.3604$ | − | − |

BGGz | $0.0117$ | $0.0294$ | $0.7428$ | $0.2621$ | $0.5984$ | − | − |

BBUXE | $0.0122$ | − | $0.3855$ | $0.1362$ | $0.3101$ | − | − |

BEWGz | $0.4117$ | $0.0795$ | $0.5477$ | $0.1917$ | $0.4446$ | $0.0050$ | $1.3587$ |

BGuGz | $0.0092$ | $0.0473$ | $0.5784$ | $0.2044$ | $0.4756$ | $2.2784$ | − |

BEMWEx | $85.9183$ | $4.5057$ | $0.1673$ | $0.0613$ | $0.1391$ | $0.0254$ | − |

BWE | $0.0251$ | − | $0.1351$ | $0.3024$ | $0.2650$ | − | − |

BEIFWE | − | − | $1.4907$ | $1.7412$ | $4.0235$ | $0.6704$ | $0.0532$ |

Model | −L | AIC | CAIC | BIC | HQIC |
---|---|---|---|---|---|

BW | $346.0174$ | $700.0102$ | $701.3145$ | $706.4336$ | $702.2892$ |

BGPW | $344.8012$ | $697.5412$ | $698.8110$ | $703.9036$ | $699.8124$ |

BGz | $303.4996$ | $614.9220$ | $616.2036$ | $621.4336$ | $617.2302$ |

BBUXGz | $301.1889$ | $612.3892$ | $614.3302$ | $620.5289$ | $615.2447$ |

BGE | $299.9142$ | $607.7419$ | $608.8894$ | $614.2301$ | $609.9163$ |

MOBE | $298.9362$ | $607.9303$ | $609.8102$ | $615.9102$ | $610.7330$ |

BEW | $298.9336$ | $607.9396$ | $609.8396$ | $615.8793$ | $610.7399$ |

BGuE | $297.8028$ | $605.5696$ | $607.5102$ | $613.6426$ | $608.4036$ |

BGLFR | $296.8389$ | $603.7339$ | $605.6396$ | $611.6896$ | $606.5012$ |

BGGz | $294.9170$ | $599.8145$ | $601.7163$ | $607.9017$ | $602.7147$ |

BBUXE | $294.8127$ | $597.6223$ | $598.9336$ | $604.0427$ | $599.9744$ |

BEWGz | $294.6036$ | $603.2112$ | $607.1745$ | $614.5107$ | $607.2338$ |

BGuGz | $294.2397$ | $600.5336$ | $603.3202$ | $610.1230$ | $603.9336$ |

BEMWEx | $294.0745$ | $600.3396$ | $603.1032$ | $609.9325$ | $603.7703$ |

BWE | $291.1437$ | $592.3103$ | $594.2147$ | $600.3223$ | $595.1196$ |

BEIFWE | $285.8012$ | $581.6302$ | $583.5415$ | $589.6520$ | $584.4415$ |

Model | $\widehat{\mathit{a}}$ | $\widehat{\mathit{b}}$ | $\widehat{{\mathit{\lambda}}_{1}}$ | $\widehat{{\mathit{\lambda}}_{2}}$ | $\widehat{{\mathit{\lambda}}_{3}}$ | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ |
---|---|---|---|---|---|---|---|

BW | $0.0391$ | − | $0.2003$ | $0.2383$ | $0.3387$ | − | − |

BGPW | $0.0292$ | − | $1.5591$ | $1.8581$ | $3.7189$ | − | − |

BE | − | − | $0.0023$ | $0.0019$ | $0.0053$ | − | − |

BGE | $0.0144$ | − | $2.4544$ | $2.8803$ | $6.0641$ | − | − |

BEW | $0.5201$ | $0.3254$ | $30.1383$ | $24.1350$ | $61.8048$ | − | − |

BGuE | $6.3113$ | $10.5332$ | $3.0661$ | $4.4849$ | $8.0431$ | − | − |

BGLFR | $6.99\times {10}^{-5}$ | $0.0011$ | $0.4174$ | $0.4862$ | $1.0193$ | − | − |

BBUXE | $0.003$ | − | $0.3622$ | $0.4241$ | $0.9071$ | − | − |

BEIFWE | − | − | $0.3625$ | $0.3412$ | $0.4125$ | $2.0365$ | $0.0789$ |

Model | −L | AIC | CAIC | BIC | HQIC |
---|---|---|---|---|---|

BW | $422.9506$ | $853.9012$ | $856.9781$ | $857.46269$ | $854.3923$ |

BGPW | $431.7917$ | $871.5834$ | $874.6603$ | $875.14489$ | $872.0745$ |

BE | $355.7320$ | $717.4642$ | $719.1785$ | $720.1353$ | $717.8325$ |

BGE | $335.2297$ | $678.4593$ | $681.5362$ | $682.0208$ | $678.9504$ |

BEW | $339.2717$ | $688.5434$ | $693.5434$ | $692.9953$ | $689.1573$ |

BGuE | $334.6306$ | $679.2612$ | $684.2612$ | $683.7131$ | $679.8751$ |

BGLFR | $331.7717$ | $673.5433$ | $678.5433$ | $677.9952$ | $674.1572$ |

BBUXE | $329.7607$ | $667.5214$ | $670.5983$ | $671.0829$ | $668.0125$ |

BEIFWE | $327.1826$ | $664.3652$ | $669.3652$ | $668.8171$ | $664.9791$ |

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## Share and Cite

**MDPI and ACS Style**

El-Morshedy, M.; Eliwa, M.S.; Tahir, M.H.; Alizadeh, M.; El-Desokey, R.; Al-Bossly, A.; Alqifari, H.
A Bivariate Extension to Exponentiated Inverse Flexible Weibull Distribution: Shock Model, Features, and Inference to Model Asymmetric Data. *Symmetry* **2023**, *15*, 411.
https://doi.org/10.3390/sym15020411

**AMA Style**

El-Morshedy M, Eliwa MS, Tahir MH, Alizadeh M, El-Desokey R, Al-Bossly A, Alqifari H.
A Bivariate Extension to Exponentiated Inverse Flexible Weibull Distribution: Shock Model, Features, and Inference to Model Asymmetric Data. *Symmetry*. 2023; 15(2):411.
https://doi.org/10.3390/sym15020411

**Chicago/Turabian Style**

El-Morshedy, Mahmoud, Mohamed S. Eliwa, Muhammad H. Tahir, Morad Alizadeh, Rana El-Desokey, Afrah Al-Bossly, and Hana Alqifari.
2023. "A Bivariate Extension to Exponentiated Inverse Flexible Weibull Distribution: Shock Model, Features, and Inference to Model Asymmetric Data" *Symmetry* 15, no. 2: 411.
https://doi.org/10.3390/sym15020411