# An Additive Chen Distribution with Applications to Lifetime Data

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## Abstract

**:**

## 1. Introduction

## 2. The New Lifetime Distribution

## 3. Measures of Central Tendency

#### 3.1. Quantile

#### 3.2. Mode

## 4. Moments and Incomplete Moments

#### 4.1. Moments

#### 4.2. Incomplete Moments

## 5. Order Statistics

## 6. Mean Residual Lifetime

## 7. Entropy

## 8. Stress-Strength Reliability

## 9. Maximum Likelihood Estimator

## 10. Case of Study

#### 10.1. Case Study 1: Lifetime of 50 Devices

#### 10.2. Case of Study 2: Field-Tracking Study of a Larger System

#### 10.3. Case of Study 3: Lifetime of 18 Electronic Devices

## 11. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. AddC Elements of the Fisher Matrix

## References

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**Figure 3.**TTT plot for Data presented in Table 2.

**Figure 4.**Reliability Plots for Lifetime data presented in Table 2.

**Figure 5.**TTT plot for Data presented in Table 4.

**Figure 6.**Reliability Plots for Lifetime data presented in Table 4.

**Figure 7.**TTT plot for Data presented in Table 6.

**Figure 8.**Reliability Plots for Lifetime data presented in Table 6.

Model | $\mathit{h}\left(\mathit{x}\right)$ |
---|---|

ACW | $\alpha \beta {x}^{\beta -1}{e}^{{x}^{\beta}}+\lambda \theta {x}^{\theta -1}$ |

APW | $\frac{\alpha \lambda {e}^{\lambda x}}{1+\alpha {e}^{\lambda x}}+\theta \beta x{}^{\beta}-1$ |

ADW | $\alpha \lambda {x}^{\lambda -1}+\beta \theta {x}^{\theta -1}$ |

Perks4 | $\frac{\theta +{e}^{\left(\right)}}{}$ |

MW | $\beta (\alpha +\lambda t){x}^{\alpha -1}{e}^{\lambda x}$ |

SZMW | $\alpha +\beta \lambda {x}^{\lambda -1}$ |

DATA | |||||||||
---|---|---|---|---|---|---|---|---|---|

0.1 | 0.2 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 6 |

7 | 11 | 12 | 18 | 18 | 18 | 18 | 18 | 21 | 32 |

36 | 40 | 45 | 46 | 47 | 50 | 55 | 60 | 63 | 63 |

67 | 67 | 67 | 67 | 72 | 75 | 79 | 82 | 82 | 83 |

84 | 84 | 84 | 85 | 85 | 85 | 85 | 85 | 86 | 86 |

**Table 3.**Estimated Values, standard errors in brackets and Statistics metrics for the Case of Study 1.

Parameters | Statistics | ||||||||
---|---|---|---|---|---|---|---|---|---|

Model | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\lambda}$ | $\mathit{\theta}$ | Loglik | AIC | BIC | K-S | p-Value |

AddC | $5.977\times {10}^{-2}(5.958\times {10}^{-4})$ | $0.249(7.114\times {10}^{-3})$ | $2.082\times {10}^{-17}(1.146\times {10}^{-22})$ | $0.822(1.654\times {10}^{-2})$ | $-203.054$ | $414.108$ | $421.756$ | $0.060$ | $0.975$ |

ACW | $4.215\times {10}^{-2}(1.021\times {10}^{-5})$ | $0.278(2.224\times {10}^{-2})$ | $1.1183\times {10}^{-2}(3.995\times {10}^{-4})$ | $86.231\left(2.414\right)$ | $-205.350$ | $418.710$ | $426.360$ | $0.070$ | $0.965$ |

APW | $7.158\times {10}^{-17}(1.112\times {10}^{-19})$ | $0.688(1.025\times {10}^{-2})$ | $0.443(1.921\times {10}^{-2})$ | $5.320\times {10}^{-2}(4.221\times {10}^{-2})$ | $-212.870$ | $433.750$ | $441.440$ | $0.091$ | $0.804$ |

ADW | $8.448\times {10}^{-9}(1.224\times {10}^{-10})$ | $0.091(3.814\times {10}^{-2})$ | $4.279(4.911\times {10}^{-2})$ | $0.466(9.814\times {10}^{-2})$ | $-221.350$ | $450.712$ | $458.360$ | $0.127$ | $0.393$ |

Perks4 | $-71.432\left(9.124\right)$ | $0.839\left(0.111\right)$ | $-6.211\times {10}^{-2}(1.321\times {10}^{-4})$ | $1.534\times {10}^{-2}(1.194\times {10}^{-3})$ | $-205.610$ | $419.112$ | $424.824$ | $0.079$ | $0.901$ |

MW | $0.355\left(0.115\right)$ | $6.221\times {10}^{-2}(2.701\times {10}^{-2})$ | $2.311\times {10}^{-2}(5\times {10}^{-3})$ | − | $-227.150$ | $460.310$ | $464.045$ | $0.134$ | $0.334$ |

SZMW | $1.311\times {10}^{-2}(3.014\times {10}^{-3})$ | $3.808\times {10}^{-9}(6.805\times {10}^{-10})$ | $4.405\left(0.145\right)$ | − | $-229.410$ | $464.821$ | $470.560$ | $0.151$ | $0.202$ |

DATA | |||||||||
---|---|---|---|---|---|---|---|---|---|

275 | 13 | 147 | 23 | 181 | 30 | 65 | 10 | 300 | 173 |

106 | 300 | 300 | 212 | 300 | 300 | 300 | 2 | 261 | 293 |

88 | 247 | 28 | 143 | 300 | 23 | 300 | 80 | 245 | 266 |

**Table 5.**Estimated Values, standard errors in brackets and Statistics metrics for the Case of Study 2.

Parameters | Statistics | ||||||||
---|---|---|---|---|---|---|---|---|---|

Model | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\lambda}$ | $\mathit{\theta}$ | Loglik | AIC | BIC | K-S | p-Value |

AddC | $3.042\times {10}^{-11}(5.958\times {10}^{-13})$ | $0.561(7.114\times {10}^{-3})$ | $0.407(1.146\times {10}^{-2})$ | $8.419\times {10}^{-2}(1.654\times {10}^{-3})$ | $-147.887$ | $303.774$ | $311.422$ | $0.112$ | $0.714$ |

ACW | $1.518\times {10}^{-2}(2.101\times {10}^{-3})$ | $0.260(3.014\times {10}^{-3})$ | $3.331\times {10}^{-3}(1.741\times {10}^{-5})$ | $259.427\left(14.558\right)$ | $-151.340$ | $310.670$ | $316.280$ | $0.132$ | $0.652$ |

APW | $5.142\times {10}^{-12}(8.063\times {10}^{-8})$ | $0.807\left(0.172\right)$ | $8.802\times {10}^{-2}(2.114\times {10}^{-3})$ | $1.114\times {10}^{-2}(9.332\times {10}^{-3})$ | $-167.910$ | $343.820$ | $349.420$ | $0.134$ | $0.655$ |

ADW | $6.915\times {10}^{-9}(9.867\times {10}^{-10})$ | $1.803\times {10}^{-2}(1.788\times {10}^{-2})$ | $3.349(5.422\times {10}^{-2})$ | $0.642\left(0.196\right)$ | $-176.97$ | $361.940$ | $367.540$ | $0.163$ | $0.401$ |

Perks4 | $-95.721\left(11.665\right)$ | $0.579\left(0.111\right)$ | $-2.421\times {10}^{-3}(1.025\times {10}^{-4})$ | $1.534\times {10}^{-2}(1.194\times {10}^{-3})$ | $-180.225$ | $368.450$ | $374.054$ | $0.185$ | $0.251$ |

MW | $0.495\left(0.228\right)$ | $6.221\times {10}^{-2}(2.701\times {10}^{-2})$ | $2.311\times {10}^{-2}(5\times {10}^{-3})$ | − | $-178.160$ | $362.330$ | $366.530$ | $0.168$ | $0.362$ |

SZMW | $3.012\times {10}^{-3}(1.025\times {10}^{-3})$ | $8.853\times {10}^{-9}(2.955\times {10}^{-9})$ | $3.266(6.241\times {10}^{-2})$ | − | $-178.470$ | $362.940$ | $367.140$ | $0.175$ | $0.316$ |

**Table 6.**Wang [7] Data of 18 electronic devices.

Data | ||||||||
---|---|---|---|---|---|---|---|---|

5 | 11 | 21 | 31 | 46 | 75 | 98 | 122 | 145 |

165 | 196 | 224 | 245 | 293 | 321 | 330 | 350 | 420 |

**Table 7.**Estimated Values, standard errors in brackets and Statistics metrics for the Case of Study 3.

Parameters | Statistics | ||||||||
---|---|---|---|---|---|---|---|---|---|

Model | $\mathit{\alpha}$ | $\mathit{\beta}$ | $\mathit{\lambda}$ | $\mathit{\theta}$ | Loglik | AIC | BIC | K-S | p-Value |

AddC | $1.946\times {10}^{-7}(1.642\times {10}^{-6})$ | $0.460(8.594\times {10}^{-2})$ | $1.576\times {10}^{-2}(1.266\times {10}^{-2})$ | $0.263(3.619\times {10}^{-2})$ | $-107.584$ | $223.168$ | $226.729$ | $0.045$ | $0.997$ |

ACW | $6.884\times {10}^{-2}(3.225\times {10}^{-3})$ | $0.228(5.332\times {10}^{-2})$ | $4.638\times {10}^{-2}(6.452\times {10}^{-3})$ | $0.224(1.445\times {10}^{-2})$ | $-116.738$ | $241.477$ | $245.039$ | $0.378$ | $0.325$ |

APW | $1.012\times {10}^{-4}(2.014\times {10}^{-3})$ | $0.799\left(0.325\right)$ | $2.722\times {10}^{-2}(4.321\times {10}^{-2})$ | $1.322\times {10}^{-2}\left(1.722\right)$ | $-108.201$ | $224.402$ | $227.963$ | $0.059$ | $0.994$ |

ADW | $4.361\times {10}^{-7}(4.058\times {10}^{-6})$ | $1.522\times {10}^{-2}(2.145\times {10}^{-2})$ | $4.025\times {10}^{-3}(2.025\times {10}^{-3})$ | $2.557\left(0.545\right)$ | $-108.880$ | $225.760$ | $229.320$ | $0.092$ | $0.991$ |

Perks4 | $-9.282\left(2.225\right)$ | $1.441\times {10}^{-2}(2.225\times {10}^{-2})$ | $-0.422(1.022\times {10}^{-2})$ | $2.821\times {10}^{-3}(3.221\times {10}^{-5})$ | $-108.125$ | $224.250$ | $227.811$ | $0.095$ | $0.995$ |

MW | $0.646\left(0.309\right)$ | $1.522\times {10}^{-2}(2.012\times {10}^{-2})$ | $4.012\times {10}^{-3}(2.114\times {10}^{-3})$ | − | $-108.935$ | $223.860$ | $226.540$ | $0.094$ | $0.992$ |

SZMW | $2.311\times {10}^{-3}(7.012\times {10}^{-3})$ | $6.211\times {10}^{-4}(5.114\times {10}^{-3})$ | $1.306\left(1.113\right)$ | − | $-110.340$ | $226.680$ | $229.350$ | $0.107$ | $0.971$ |

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**MDPI and ACS Style**

Méndez-González, L.C.; Rodríguez-Picón, L.A.; Pérez-Olguín, I.J.C.; Vidal Portilla, L.R.
An Additive Chen Distribution with Applications to Lifetime Data. *Axioms* **2023**, *12*, 118.
https://doi.org/10.3390/axioms12020118

**AMA Style**

Méndez-González LC, Rodríguez-Picón LA, Pérez-Olguín IJC, Vidal Portilla LR.
An Additive Chen Distribution with Applications to Lifetime Data. *Axioms*. 2023; 12(2):118.
https://doi.org/10.3390/axioms12020118

**Chicago/Turabian Style**

Méndez-González, Luis Carlos, Luis Alberto Rodríguez-Picón, Ivan Juan Carlos Pérez-Olguín, and Luis Ricardo Vidal Portilla.
2023. "An Additive Chen Distribution with Applications to Lifetime Data" *Axioms* 12, no. 2: 118.
https://doi.org/10.3390/axioms12020118