# Bivariate Discrete Odd Generalized Exponential Generator of Distributions for Count Data: Copula Technique, Mathematical Theory, and Applications

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## Abstract

**:**

## 1. Introduction

## 2. The BDOGE-G Class

## 3. Distributional Properties

#### 3.1. Median Correlation Coefficient

#### 3.2. The Conditional CDF of ${X}_{1}$ Given ${X}_{2}={x}_{2}$ (${X}_{2}\le {x}_{2}$)

#### 3.3. The Conditional Expectation of ${X}_{1}$ Given ${X}_{2}={x}_{2}$

## 4. The BDsOGE-Weibull (BDsOGEW) Distribution

## 5. Point and Interval Estimations

#### 5.1. Maximum Likelihood Estimation (MLE)

#### 5.2. Asymptotic Confidence Intervals

## 6. Simulation: Estimators Performance

- Scheme I: (∀$p$= $0.1,\nu $=$0.3,\gamma $=$0.5,$ $a$=$0.2$ | ${n}_{1}$= 20, ${n}_{2}$= 50, ${n}_{3}$= 150, ${n}_{4}$= 300, ${n}_{5}$= 500, ${n}_{6}$= 700);
- Scheme II: (∀$p$= $0.2,\nu $=$0.5,\gamma $=$0.3,$ $a$=$0.5$ | ${n}_{1}$= 20, ${n}_{2}$= 50, ${n}_{3}$= 150, ${n}_{4}$= 300, ${n}_{5}$= 500, ${n}_{6}$= 700);
- Scheme III: (∀$p$= $0.5,\nu $=$0.3,\gamma $=$0.7,$ $a$=$0.9$ | ${n}_{1}$= 20, ${n}_{2}$= 50, ${n}_{3}$= 150, ${n}_{4}$= 300, ${n}_{5}$= 500, ${n}_{6}$= 700)

## 7. Data Analysis

#### 7.1. Data Set I: Nasal Drainage Severity Score

#### 7.2. Data Set II: Football Score

## 8. Concluding Remarks and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- The PMF of the BDsOGE-G

- Estimation code for real data

## References

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**Figure 14.**A 3-dimensional surface profile likelihood of a, $\gamma $, p and v based on data set II.

${n}_{1}=20$ | ${n}_{2}=50$ | ||||||||

p | $\nu $ | $\gamma $ | $a$ | p | $\nu $ | $\gamma $ | $a$ | ||

Bias | $0.10125$ | $0.21458$ | $0.18256$ | $0.25203$ | $0.05236$ | $0.18256$ | $0.12469$ | $0.19636$ | |

MSE | $0.09856$ | $0.18256$ | $0.15246$ | $0.21552$ | $0.04254$ | $0.15635$ | $0.09850$ | $0.16504$ | |

CP | $0.88719$ | $0.92335$ | $0.93419$ | $0.83963$ | $0.88436$ | $0.92144$ | $0.93223$ | $0.83745$ | |

$95\%$ CI${}_{LB}$ | $0.06523$ | $0.24714$ | $0.42302$ | $0.14025$ | $0.07136$ | $0.25748$ | $0.44120$ | $0.15699$ | |

$95\%$ CI${}_{UB}$ | $0.14569$ | $0.35125$ | $0.59368$ | $0.27145$ | $0.13825$ | $0.34283$ | $0.56774$ | $0.26632$ | |

${n}_{3}=150$ | ${n}_{4}=300$ | ||||||||

p | $\nu $ | $\gamma $ | $a$ | p | $\nu $ | $\gamma $ | $a$ | ||

Bias | $0.01365$ | $0.12145$ | $0.08254$ | $0.14231$ | $0.00858$ | $0.07156$ | $0.0236$ | $0.11258$ | |

MSE | $0.01258$ | $0.10395$ | $0.06632$ | $0.12569$ | $0.00652$ | $0.05236$ | $0.01428$ | $0.09254$ | |

CP | $0.88253$ | $0.92074$ | $0.93064$ | $0.83164$ | $0.88162$ | $0.91886$ | $0.92846$ | $0.82854$ | |

$95\%$ CI${}_{LB}$ | $0.07512$ | $0.26314$ | $0.46256$ | $0.16369$ | $0.08124$ | $0.27142$ | $0.47428$ | $0.17402$ | |

$95\%$ CI${}_{UB}$ | $0.12415$ | $0.34472$ | $0.54239$ | $0.24012$ | $0.11932$ | $0.33748$ | $0.53201$ | $0.23148$ | |

${n}_{5}=500$ | ${n}_{6}=700$ | ||||||||

p | $\nu $ | $\gamma $ | $a$ | p | $\nu $ | $\gamma $ | $a$ | ||

Bias | $0.00103$ | $0.01254$ | $0.00125$ | $0.06317$ | $0.00003$ | $0.00235$ | $0.00024$ | $0.00301$ | |

MSE | $0.00084$ | $0.00825$ | $0.00052$ | $0.01426$ | $0.00001$ | $0.00082$ | $0.00011$ | $0.00082$ | |

CP | $0.87949$ | $0.91278$ | $0.92676$ | $0.81746$ | $0.87552$ | $0.91075$ | $0.92019$ | $0.81298$ | |

$95\%$ CI${}_{LB}$ | $0.08412$ | $0.28301$ | $0.48241$ | $0.18012$ | $0.09174$ | $0.28102$ | $0.49102$ | $0.19125$ | |

$95\%$ CI${}_{UB}$ | $0.11363$ | $0.32823$ | $0.53294$ | $0.21989$ | $0.10857$ | $0.31025$ | $0.52138$ | $0.21011$ |

${n}_{1}=20$ | ${n}_{2}=50$ | ||||||||

p | $\nu $ | $\gamma $ | $a$ | p | $\nu $ | $\gamma $ | $a$ | ||

Bias | $0.12369$ | $0.23529$ | $0.13085$ | $0.16625$ | $0.10238$ | $0.20136$ | $0.09820$ | $0.14823$ | |

MSE | $0.10925$ | $0.22968$ | $0.12821$ | $0.15530$ | $0.09569$ | $0.18014$ | $0.07742$ | $0.13224$ | |

CP | $0.91203$ | $0.82734$ | $0.94537$ | $0.85016$ | $0.91027$ | $0.82256$ | $0.94521$ | $0.84771$ | |

$95\%$ CI${}_{LB}$ | $0.15365$ | $0.40123$ | $0.22303$ | $0.46369$ | $0.16325$ | $0.42013$ | $0.24120$ | $0.47102$ | |

$95\%$ CI${}_{UB}$ | $0.26636$ | $0.59325$ | $0.38204$ | $0.54215$ | $0.24932$ | $0.57012$ | $0.36025$ | $0.53325$ | |

${n}_{3}=150$ | ${n}_{4}=300$ | ||||||||

p | $\nu $ | $\gamma $ | $a$ | p | $\nu $ | $\gamma $ | $a$ | ||

Bias | $0.08256$ | $0.16328$ | $0.04230$ | $0.12012$ | $0.02137$ | $0.11205$ | $0.00803$ | $0.08825$ | |

MSE | $0.06636$ | $0.14200$ | $0.03047$ | $0.09825$ | $0.01452$ | $0.09852$ | $0.00625$ | $0.06328$ | |

CP | $0.90734$ | $0.82019$ | $0.94338$ | $0.84309$ | $0.90193$ | $0.81578$ | $0.94219$ | $0.84290$ | |

$95\%$ CI${}_{LB}$ | $0.17714$ | $0.43025$ | $0.25202$ | $0.48158$ | $0.18230$ | $0.45201$ | $0.28323$ | $0.48623$ | |

$95\%$ CI${}_{UB}$ | $0.23852$ | $0.55236$ | $0.34525$ | $0.52745$ | $0.22825$ | $0.54414$ | $0.32256$ | $0.51445$ | |

${n}_{5}=500$ | ${n}_{6}=700$ | ||||||||

p | $\nu $ | $\gamma $ | $a$ | p | $\nu $ | $\gamma $ | $a$ | ||

Bias | $0.00615$ | $0.02167$ | $0.00012$ | $0.00215$ | $0.00023$ | $0.00273$ | $0.00003$ | $0.00052$ | |

MSE | $0.00321$ | $0.00825$ | $0.00008$ | $0.00102$ | $0.00004$ | $0.00019$ | $0.00001$ | $0.00008$ | |

CP | $0.89723$ | $0.81426$ | $0.94139$ | $0.84311$ | $0.89227$ | $0.81337$ | $0.93772$ | $0.84382$ | |

$95\%$ CI${}_{LB}$ | $0.18936$ | $0.46236$ | $0.29201$ | $0.49012$ | $0.19102$ | $0.47525$ | $0.29025$ | $0.49523$ | |

$95\%$ CI${}_{UB}$ | $0.21985$ | $0.53926$ | $0.30941$ | $0.51125$ | $0.20824$ | $0.53101$ | $0.30125$ | $0.50585$ |

${n}_{1}=20$ | ${n}_{2}=50$ | ||||||||

p | $\nu $ | $\gamma $ | $a$ | p | $\nu $ | $\gamma $ | $a$ | ||

Bias | $0.09858$ | $0.24223$ | $0.19302$ | $0.16014$ | $0.07145$ | $0.19073$ | $0.16325$ | $0.14085$ | |

MSE | $0.07145$ | $0.21014$ | $0.16328$ | $0.14025$ | $0.05241$ | $0.17452$ | $0.14145$ | $0.13748$ | |

CP | $0.92747$ | $0.81457$ | $0.97194$ | $0.95712$ | $0.91265$ | $0.82602$ | $0.97188$ | $0.95639$ | |

$95\%$ CI${}_{LB}$ | $0.46568$ | $0.19365$ | $0.58636$ | $0.75145$ | $0.47188$ | $0.22325$ | $0.61258$ | $0.79569$ | |

$95\%$ CI${}_{UB}$ | $0.54258$ | $0.41256$ | $0.82414$ | $1.15369$ | $0.53521$ | $0.38525$ | $0.79201$ | $1.10254$ | |

${n}_{3}=150$ | ${n}_{4}=300$ | ||||||||

p | $\nu $ | $\gamma $ | $a$ | p | $\nu $ | $\gamma $ | $a$ | ||

Bias | $0.02035$ | $0.14625$ | $0.13205$ | $0.12638$ | $0.00413$ | $0.12025$ | $0.11015$ | $0.11365$ | |

MSE | $0.01996$ | $0.13236$ | $0.11859$ | $0.12025$ | $0.00251$ | $0.11301$ | $0.09852$ | $0.09852$ | |

CP | $0.91746$ | $0.82746$ | $0.96984$ | $0.95612$ | $0.92957$ | $0.88230$ | $0.96901$ | $0.95598$ | |

$95\%$ CI${}_{LB}$ | $0.48194$ | $0.24189$ | $0.64302$ | $0.82638$ | $0.48831$ | $0.25885$ | $0.66358$ | $0.86677$ | |

$95\%$ CI${}_{UB}$ | $0.52825$ | $0.34125$ | $0.76012$ | $0.98510$ | $0.51254$ | $0.33236$ | $0.73258$ | $0.93748$ | |

${n}_{5}=500$ | ${n}_{6}=700$ | ||||||||

p | $\nu $ | $\gamma $ | $a$ | p | $\nu $ | $\gamma $ | $a$ | ||

Bias | $0.00074$ | $0.08236$ | $0.08858$ | $0.09014$ | $0.00002$ | $0.00714$ | $0.00752$ | $0.01173$ | |

MSE | $0.00066$ | $0.06326$ | $0.07968$ | $0.06852$ | $0.00001$ | $0.00513$ | $0.00524$ | $0.00721$ | |

CP | $0.90166$ | $0.88109$ | $0.96836$ | $0.95583$ | $0.90654$ | $0.88112$ | $0.96554$ | $0.95566$ | |

$95\%$ CI${}_{LB}$ | $0.49134$ | $0.27256$ | $0.68225$ | $0.87480$ | $0.49014$ | $0.28541$ | $0.69221$ | $0.88369$ | |

$95\%$ CI${}_{UB}$ | $0.51015$ | $0.32748$ | $0.71529$ | $0.93254$ | $0.50895$ | $0.31526$ | $0.70418$ | $0.91596$ |

Model | MLEs | $-\mathit{l}$ | AIC | HQIC |
---|---|---|---|---|

BDsE | ${\widehat{\nu}}_{1}=0.846,{\widehat{\nu}}_{2}=0.792,{\widehat{\nu}}_{3}=0.693$ | $88.002$ | $182.004$ | $183.349$ |

BPo-4P | ${\widehat{\lambda}}_{1}=0.262,{\widehat{\gamma}}_{1}=0.165,{\widehat{\lambda}}_{2}=0.405,{\widehat{\gamma}}_{2}=2.971$ | $77.664$ | $163.328$ | $165.121$ |

IBPo | ${\widehat{\lambda}}_{1}=1.499,{\widehat{\lambda}}_{2}=1.367$ | $92.478$ | $188.956$ | $189.853$ |

BDsIE | ${\widehat{\nu}}_{1}=0.501,{\widehat{\nu}}_{2}=0.622,{\widehat{\nu}}_{3}=0.383$ | $92.482$ | $190.964$ | $192.309$ |

BDsIR | ${\widehat{\nu}}_{1}=0.262,{\widehat{\nu}}_{2}=0.405,{\widehat{\nu}}_{3}=0.363$ | $78.659$ | $163.318$ | $164.663$ |

BDsIW | ${\widehat{\nu}}_{1}=0.192,{\widehat{\nu}}_{2}=0.337,{\widehat{\nu}}_{3}=0.360,\widehat{\zeta}=2.453$ | $76.513$ | $161.026$ | $162.815$ |

BDsOGEW | $\widehat{\gamma}=0.574,\widehat{\nu}=2.457,\widehat{p}=0.722,\widehat{a}=0.780$ | $71.162$ | $150.324$ | $152.117$ |

Model | MLEs | $-\mathit{l}$ | AIC | HQIC |
---|---|---|---|---|

BDsE | ${\widehat{\nu}}_{1}=0.625,{\widehat{\nu}}_{2}=0.812,{\widehat{\nu}}_{3}=0.713$ | $75.421$ | $156.842$ | $157.929$ |

BDsR | ${\widehat{\nu}}_{1}=0.790,{\widehat{\nu}}_{2}=0.872,{\widehat{\nu}}_{3}=0.905$ | $63.931$ | $133.862$ | $134.949$ |

BDsW | ${\widehat{\nu}}_{1}=1.36,{\widehat{\nu}}_{2}=2.10,{\widehat{\nu}}_{3}=2.27,\widehat{\zeta}=2.125$ | $63.911$ | $135.822$ | $137.271$ |

BPo${}_{min}$ | ${\widehat{\nu}}_{1}=1.36,{\widehat{\nu}}_{2}=2.10,{\widehat{\nu}}_{3}=2.27$ | $64.228$ | $134.456$ | $135.543$ |

BPo-3P | ${\widehat{\gamma}}_{1}=1.08,{\widehat{\gamma}}_{2}=1.38,{\widehat{\gamma}}_{3}=0.70$ | $64.932$ | $135.864$ | $136.951$ |

IBPo | ${\widehat{\lambda}}_{1}=1.08,{\widehat{\lambda}}_{2}=1.38$ | $67.623$ | $139.246$ | $139.971$ |

BDsIE | ${\widehat{\nu}}_{1}=0.669,{\widehat{\nu}}_{2}=0.388,{\widehat{\nu}}_{3}=0.514$ | $78.541$ | $163.082$ | $164.169$ |

BDsIR | ${\widehat{\nu}}_{1}=0.493,{\widehat{\nu}}_{2}=0.212,{\widehat{\nu}}_{3}=0.561$ | $64.102$ | $134.204$ | $135.291$ |

BDsOGEW | $\widehat{\gamma}=0.759,\widehat{\nu}=23.807,\widehat{p}=0.213,\widehat{a}=0.399$ | $62.589$ | $133.178$ | $134.627$ |

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## Share and Cite

**MDPI and ACS Style**

Al-Essa, L.A.; Eliwa, M.S.; Shahen, H.S.; Khalil, A.A.; Alqifari, H.N.; El-Morshedy, M.
Bivariate Discrete Odd Generalized Exponential Generator of Distributions for Count Data: Copula Technique, Mathematical Theory, and Applications. *Axioms* **2023**, *12*, 534.
https://doi.org/10.3390/axioms12060534

**AMA Style**

Al-Essa LA, Eliwa MS, Shahen HS, Khalil AA, Alqifari HN, El-Morshedy M.
Bivariate Discrete Odd Generalized Exponential Generator of Distributions for Count Data: Copula Technique, Mathematical Theory, and Applications. *Axioms*. 2023; 12(6):534.
https://doi.org/10.3390/axioms12060534

**Chicago/Turabian Style**

Al-Essa, Laila A., Mohamed S. Eliwa, Hend S. Shahen, Amal A. Khalil, Hana N. Alqifari, and Mahmoud El-Morshedy.
2023. "Bivariate Discrete Odd Generalized Exponential Generator of Distributions for Count Data: Copula Technique, Mathematical Theory, and Applications" *Axioms* 12, no. 6: 534.
https://doi.org/10.3390/axioms12060534