# Bivariate Generalized Half-Logistic Distribution: Properties and Its Application in Household Financial Affordability in KSA

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## Abstract

**:**

## 1. Introduction

**Definition 1.**

**Definition 2.**

## 2. FGM Bivariate Generalized Half-Logistic Distribution (FGMBGHLD)

## 3. Statistical Properties of the FGMBGHLD

#### 3.1. Marginal PDFs

#### 3.2. Moment Generating Function

#### 3.3. Product Moments

#### 3.4. Reliability and Hazard Rate Functions

## 4. Reliability for Dependence Stress–Strength Model

## 5. Estimation Method for the Distribution Parameters

## 6. Estimation of the Stress–Strength Distribution Parameter

#### 6.1. Maximum Likelihood Estimate of R

#### 6.2. Asymptotic Confidence Interval (ACI)

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Proof.**

## 7. Simulation

#### 7.1. Random Variate Generation

- (i)
- Generate u and v independently from a uniform $(0,1)$ distribution.
- (ii)
- Put ${y}_{1}={\sigma}_{1}\phantom{\rule{4pt}{0ex}}\mathrm{ln}(1-2{\phantom{\rule{4pt}{0ex}}(1-u)}^{{\mu}_{1}})\phantom{\rule{4pt}{0ex}}$.
- (iii)
- Put $F\left({y}_{2}\left|{y}_{1}\right.\right)=v$ to find ${y}_{2}$ using numerical simulation.
- (iv)
- Repeat (i) to (iii) n-times to obtain ${(y}_{1j},{y}_{2j})$, $j=1,\dots ,n.$

#### 7.2. Bootstrap Confidence Interval (BCI)

- (i)
- Select the simple random sample $\left({x}_{i},{y}_{i}\right)$, $i=1,\dots ,n.$
- (ii)
- Re-sample the simple random sample $\left({x}_{i},{y}_{i}\right)$ with replacement.
- (iii)
- Obtain the new simple random sample $({x}_{i}^{*},{y}_{i}^{*})$.
- (iv)
- Compute ${R}^{*}.$
- (v)
- Repeat step (i)–(iv) B-times and compute ${R}_{1}^{*},\dots ,{R}_{n}^{*}$.
- (vi)
- Arrange ${R}_{1}^{*},\dots ,{R}_{n}^{*}$, from the smallest to the largest ${R}_{\left(1\right)}^{*},\dots ,{R}_{\left(n\right)}^{*}$.
- (vii)
- Construct a $100\left(1-\alpha \right)\%$ ACI of R as$$\left({R}_{\frac{\alpha}{2},B}^{*},\phantom{\rule{4pt}{0ex}}{R}_{\left(1-\frac{\alpha}{2}\right),B}^{*}\right).$$

#### 7.3. Experiment

- 1.
- Assume some true values of the parameters ${\mu}_{1},\phantom{\rule{4pt}{0ex}}{\mu}_{2},\phantom{\rule{4pt}{0ex}}\sigma ,\phantom{\rule{4pt}{0ex}}\theta $ and compute the corresponding true values of $R.$Case 1: If ${\mu}_{1}=0.5,\phantom{\rule{4pt}{0ex}}{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mu}_{2}=1.5,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sigma =1,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta =-0.75$, then $R=0.8678.$Case 2: If ${\mu}_{1}=0.5,\phantom{\rule{4pt}{0ex}}{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mu}_{2}=1.5,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sigma =1,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta =-0.25$, then $R=0.7892.$Case 3: If ${\mu}_{1}=0.5,\phantom{\rule{4pt}{0ex}}{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mu}_{2}=1.5,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sigma =1,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta =0.25$, then $R=0.7107.$Case 4: If ${\mu}_{1}=0.5,\phantom{\rule{4pt}{0ex}}{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mu}_{2}=1.5,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sigma =1,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta =0.75$, then $R=0.6321.$
- 2.
- Use the algorithm in Section 7.2 to generate different sample sizes with $n=30,50,70$ and 100, with 10,000 replications. All computations are obtained using Mathematica 11.1.
- 3.
- Calculate ${R}_{MLE}$ according to the methodology in Section 6.1 and the “average ${R}_{MLE}$”, say ${R}_{MLE}^{*}$, based on all the samples at a fixed size.
- 4.
- Evaluate the ACI and BCI according to the methodology in Section 6.2 and Section 7.2.
- 5.
- Study the behavior of ${R}_{MLE}$ by evaluating the bias defined by the “average of $\left({R}_{MLE}-R\right)$” and the mean square error (MSE) indicated as the “average of ${\left({R}_{MLE}-R\right)}^{2}$”.
- 6.
- In the context of interval estimation, we compare the ACI and BCI using the asymptotic confidence length (ACL) and converge probability (CP).

- 1.
- At $n=100$, the value of the MSE becomes very small.
- 2.
- In general, the length of the ACI becomes smaller than the length of the BCI.
- 3.
- When the ACL decreases, the CP increases.
- 4.
- The CP in almost all cases of the ACI is more than the CP in the BCI.

## 8. Application: Household Financial Affordability in KSA 2018

**Case 1**: If X and Y are independent with X following the GHLD$({\mu}_{1},\sigma )$ and Y following the GHLD$({\mu}_{2},\sigma )$, and the dependent parameter $\theta $ is set as 0;**Case 2:**If X and Y are dependent with $(X$,$Y)$ following the FGMBGHLD.

- 1.
- Since $\theta $ is estimated as $0.4713$, and is therefore positive, then the relation between X and Y is positive, as we see in Figure 3.
- 2.
- The measure of affordability when X and Y are dependent is less than when X and Y are independent, so the case of dependent variables is more realistic.

**Case 1**: If X and Y are independent with X following the $Weibull({\alpha}_{1},\beta )$ and Y following the $Weibull({\alpha}_{2},\beta )$;**Case 2:**If X and Y are dependent with $(X$,$Y)$ following the BWD.

**Table 5.**The MLEs, ACIs, and ACLs of the BWD parameters for the income and consumption of KSA, year 2018.

Case | MLE | MLE for R | ACI | ACL |
---|---|---|---|---|

Case 1 | ${\widehat{\alpha}}_{1}=6.5$ ${\widehat{\alpha}}_{2}=7.5$ $\widehat{\beta}=1.45$ | $0.4642$ | (0.4631, 0.6461) | 0.1820 |

Case 2 | ${\widehat{\alpha}}_{1}=6.5$ ${\widehat{\alpha}}_{2}=7.5$ $\widehat{\beta}=1.45$ $\widehat{\theta}=0.1082$ | $0.4275$ | (0.3456, 0.6058) | 0.2602 |

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Three-dimensional plots for the PDF and CDF of the FGMBGHLD with different values of $\theta $ (for ${\mu}_{1}={\mu}_{2}=0.5$, ${\sigma}_{1}=0.2$ and ${\sigma}_{2}=0.1$).

**Figure 4.**The estimated PDF and CDF of the FGMBGHLD for the income and consumption of KSA, year 2018.

Sample Size | ${\mathit{R}}_{\mathbf{true}}$ | ${\mathit{R}}_{\mathbf{MLE}}^{*}$ | MLE | ACI | BCI | |||
---|---|---|---|---|---|---|---|---|

Bias | MSE | ACL | CP | ACL | CP | |||

${\mu}_{\mathbf{1}}=\mathbf{0.5},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mu}_{\mathbf{2}}=\mathbf{1.5},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sigma =\mathbf{1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta =-\mathbf{0.75}$ | ||||||||

$n=30$ | 0.8678 | 0.3960 | −0.0228 | 0.0157 | 0.205 | 0.935 | 0.598 | 0.780 |

$n=50$ | 0.6463 | −0.0064 | 0.0020 | 0.474 | 0.831 | 0.838 | 0.690 | |

$n=70$ | 0.7841 | −0.0084 | 0.0049 | 0.397 | 0.846 | 0.858 | 0.684 | |

$n=100$ | 0.3392 | −0.0037 | 0.0014 | 0.223 | 0.980 | 0.503 | 0.819 | |

${\mu}_{\mathbf{1}}=\mathbf{0.5},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mu}_{\mathbf{2}}=\mathbf{1.5},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sigma =\mathbf{1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta =-\mathbf{0.25}$ | ||||||||

$n=30$ | 0.7892 | 0.1027 | −0.0157 | 0.0074 | 0.792 | 0.856 | 0.511 | 0.818 |

$n=50$ | 0.4670 | −0.0044 | 0.0009 | 0.418 | 0.864 | 0.463 | 0.842 | |

$n=70$ | 0.1989 | −0.0011 | 0.0001 | 0.251 | 0.926 | 0.524 | 0.810 | |

$n=100$ | 0.4110 | −0.0052 | 0.0027 | 0.226 | 0.932 | 0.282 | 0.903 | |

${\mu}_{\mathbf{1}}=\mathbf{0.5},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mu}_{\mathbf{2}}=\mathbf{1.5},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sigma =\mathbf{1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta =\mathbf{0.25}$ | ||||||||

$n=30$ | 0.7107 | 0.0858 | −0.0208 | 0.0130 | 0.171 | 0.940 | 0.730 | 0.731 |

$n=50$ | 0.3216 | −0.0077 | 0.0030 | 0.367 | 0.861 | 0.806 | 0.693 | |

$n=70$ | 0.7090 | −0.0001 | 0.0001 | 0.537 | 0.980 | 0.095 | 0.970 | |

$n=100$ | 0.6757 | −0.0003 | 0.0001 | 0.099 | 0.967 | 0.126 | 0.960 | |

${\mu}_{\mathbf{1}}=\mathbf{0.5},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mu}_{\mathbf{2}}=\mathbf{1.5},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\sigma =\mathbf{1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta =\mathbf{0.75}$ | ||||||||

$n=30$ | 0.6321 | 0.6630 | 0.0010 | 0.0000 | 0.272 | 0.912 | 0.189 | 0.941 |

$n=50$ | 0.5554 | −0.0015 | 0.0001 | 0.308 | 0.894 | 0.194 | 0.922 | |

$n=70$ | 0.2641 | −0.0052 | 0.0019 | 0.291 | 0.887 | 0.136 | 0.946 | |

$n=100$ | 0.6775 | 0.0004 | 0.0000 | 0.108 | 0.965 | 0.0931 | 0.970 |

**Table 2.**Average household monthly income (X) and consumption expenditure (Y) by administrative region for Saudi households in 2018.

Administrative Region | Income | Consumption |
---|---|---|

Riyadh | 16,011 | 15,917 |

Makkah | 14,648 | 14,256 |

Madinah | 12,016 | 118,322 |

Al-Qassim | 15,322 | 14,371 |

Eastern Region | 17,872 | 17,665 |

Asir | 11,817 | 11,666 |

Tabuk | 11,024 | 10,890 |

Hail | 11,571 | 11,461 |

North Board | 12,051 | 11,876 |

Jazan | 15,199 | 15,071 |

Najran | 11,388 | 11,376 |

Al-Baha | 13,728 | 13,605 |

Aljouf | 14,193 | 14,101 |

Total | 14,823 | 14,584 |

Measure | Income | Consumption |
---|---|---|

Min | 11,024 | 10,890 |

Max | 17,872 | 17,665 |

Median | 13,728 | 13,605 |

SE | 592.605 | 574.401 |

Skewness | 0.529 | 0.637 |

Kurtosis | −0.686 | −0.378 |

Mean | 13,603.076 | 13,391.307 |

Case | MLE | MLE for R | ACI | ACL |
---|---|---|---|---|

Case 1 | ${\widehat{\mu}}_{1}=0.0143$ ${\widehat{\mu}}_{2}=0.0270$ $\widehat{\sigma}=0.3529$ | $0.3462$ | (0.2979, 0.3945) | 0.0965 |

Case 2 | ${\widehat{\mu}}_{1}=0.0135$ ${\widehat{\mu}}_{2}=0.0201$ $\widehat{\sigma}=0.1403$ $\widehat{\theta}=0.4713$ | $0.2149$ | (0.1248, 0.3050) | 0.1802 |

Distribution | p-Value |
---|---|

FGMBGHLD | 0.4999 |

BWD | 0.2067 |

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**MDPI and ACS Style**

Hassan, M.K.H.; Chesneau, C.
Bivariate Generalized Half-Logistic Distribution: Properties and Its Application in Household Financial Affordability in KSA. *Math. Comput. Appl.* **2022**, *27*, 72.
https://doi.org/10.3390/mca27040072

**AMA Style**

Hassan MKH, Chesneau C.
Bivariate Generalized Half-Logistic Distribution: Properties and Its Application in Household Financial Affordability in KSA. *Mathematical and Computational Applications*. 2022; 27(4):72.
https://doi.org/10.3390/mca27040072

**Chicago/Turabian Style**

Hassan, Marwa K. H., and Christophe Chesneau.
2022. "Bivariate Generalized Half-Logistic Distribution: Properties and Its Application in Household Financial Affordability in KSA" *Mathematical and Computational Applications* 27, no. 4: 72.
https://doi.org/10.3390/mca27040072