# Scaled-Invariant Extended Quasi-Lindley Model: Properties, Estimation, and Application

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## Abstract

**:**

## 1. Introduction

## 2. Scaled-Invariant Extended QL Model

#### Reliability Properties

**Proposition**

**1.**

**Proof.**

## 3. Estimation

#### 3.1. ML Method

#### 3.2. LSE Method

#### 3.3. Weighted LSE Method

#### 3.4. EM Algorithm

## 4. Simulations

- First, drive one random instance from a multinomial model with parameters $({p}_{1},{p}_{2},{p}_{3},n)$, where ${p}_{1}=1/(1+\alpha +{\alpha}^{2})$, ${p}_{2}=\alpha /(1+\alpha +{\alpha}^{2})$, and ${p}_{3}=1-{p}_{1}-{p}_{2}$. Assume the derived instance is $({k}_{1},{k}_{2},{k}_{3})$.
- Generate and mix three identical and independent (iid) random samples from $G(1,\xi )$, $G(2,\xi )$, and $G(3,\xi )$ with sizes ${k}_{1}$, ${k}_{2}$, and ${k}_{3}$ respectively.

## 5. Application

## 6. Conclusions

- Estimate the unknown parameters of the proposed model, along with the reliability and hazard rate functions under different types of censoring schemes, such as progressive type II, hybrid, general progressive, and adaptive censoring schemes.
- Consider the maximum likelihood and maximum product-of-spacing methods to determine the point estimates and approximate confidence intervals of the various model parameters.
- Provide Bayesian estimates based on the likelihood function and product of the distance function of the unknown parameters using the quadratic error loss function with independent gamma priors.
- The methods investigated in this study can be extended to study estimation problems in more complex cases.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Proposition**

**1.**

## Appendix B. E Step and M Step of EM Algorithm

## References

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**Figure 3.**The TTT plot (

**left**) and histogram along with estimated PDF (

**right**) of times between failures of air conditioning system. The red plus in the left figure shows calculated TTT for each data entry.

**Figure 4.**The empirical and estimated CDF for QL and some alternative models of times between failures of air conditioning system.

**Table 1.**Simulation results for ML, LSE, AD, and EM algorithm. The first and second lines of every cell correspond to $\alpha $ and $\xi $.

$\mathit{n}$ | |||||
---|---|---|---|---|---|

Method | 80 | 150 | |||

α,$\xi $ | B | MSE | B | MSE | |

MLE | 0.1, 0.1 | 0.2093 0.0221 | 0.1413 0.0015 | 0.1486 0.0166 | 0.0853 0.0010 |

0.3, 0.5 | 0.1009 0.0415 | 0.1222 0.0209 | 0.0466 0.0168 | 0.0771 0.0133 | |

0.8, 1 | 0.0598 0.0029 | 0.1716 0.0326 | 0.0273 −0.0043 | 0.0948 0.0201 | |

EM | 0.1, 0.1 | 0.0833 0.0097 | 0.0377 0.0004 | 0.0463 0.0054 | 0.0126 0.0002 |

0.3, 0.5 | 0.1346 0.0530 | 0.1095 0.0181 | 0.0807 0.0329 | 0.0554 0.0100 | |

0.8, 1 | 0.1023 0.0223 | 0.1856 0.0299 | 0.0281 0.0015 | 0.0728 0.0156 | |

LSE | 0.1, 0.1 | 0.2350 0.0306 | 0.1970 0.0026 | 0.1734 0.0230 | 0.1185 0.0016 |

0.3, 0.5 | 0.0906 0.0523 | 0.1726 0.0315 | 0.0466 0.0287 | 0.1083 0.0206 | |

0.8, 1 | 0.0609 0.0075 | 0.2960 0.0500 | 0.0143 −0.0038 | 0.1085 0.0270 | |

Weighted LSE (AD) | 0.1, 0.1 | 0.0283 0.0097 | 0.0592 0.0009 | 0.0249 0.0084 | 0.0392 0.0006 |

0.3, 0.5 | −0.1116 −0.0233 | 0.1157 0.0211 | −0.1373 −0.0367 | 0.0778 0.0121 | |

0.8, 1 | −0.3220 −0.1497 | 0.2795 0.0700 | −0.2426 −0.1230 | 0.1963 0.0466 |

**Table 2.**Time interval, in terms of hours, between successive failures of air conditioner system of Boeing 720 aircraft.

59 | 20 | 68 | 67 | 25 | 13 | 5 | 79 | 76 |

127 | 117 | 100 | 52 | 189 | 398 | 60 | 117 | 263 |

143 | 39 | 194 | 128 | 160 | 88 | 74 | 66 | 199 |

180 | 156 |

Model | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\beta}}$ | $\widehat{\mathit{\xi}}$ | AIC | CVM | AD | KS |
---|---|---|---|---|---|---|---|

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

EQL | 1.9668 | — | 0.0215 | 331.22 | 0.0278 0.9843 | 0.1833 0.9944 | 0.0801 0.9923 |

Gamma | 1.7195 | — | 0.0153 | 331.55 | 0.0363 0.9539 | 0.2399 0.9754 | 0.1028 0.9190 |

EG | 2.8250 | 0.0823 | 0.1459 | 334.57 | 0.0647 0.7882 | 0.3836 0.8638 | 0.1308 0.7037 |

LG | 1.4504 | 1.1997 | 0.0142 | 333.59 | 0.0373 0.9682 | 0.2454 0.9727 | 0.1041 0.9120 |

MOG | 1.6439 | 1.2563 | 0.0161 | 333.37 | 0.0322 0.9705 | 0.2169 0.9851 | 0.0965 0.9498 |

QL | 0.1382 | — | 0.0167 | 331.35 | 0.0320 0.9712 | 0.2057 0.9888 | 0.0965 0.9499 |

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**MDPI and ACS Style**

Kayid, M.; Abouammoh, A.; Alomani, G.
Scaled-Invariant Extended Quasi-Lindley Model: Properties, Estimation, and Application. *Symmetry* **2023**, *15*, 1780.
https://doi.org/10.3390/sym15091780

**AMA Style**

Kayid M, Abouammoh A, Alomani G.
Scaled-Invariant Extended Quasi-Lindley Model: Properties, Estimation, and Application. *Symmetry*. 2023; 15(9):1780.
https://doi.org/10.3390/sym15091780

**Chicago/Turabian Style**

Kayid, Mohamed, Abdulrahman Abouammoh, and Ghadah Alomani.
2023. "Scaled-Invariant Extended Quasi-Lindley Model: Properties, Estimation, and Application" *Symmetry* 15, no. 9: 1780.
https://doi.org/10.3390/sym15091780