# A New Insight into Reliability Data Modeling with an Exponentiated Composite Exponential-Pareto Model

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. The Composite Distributions

#### 2.2. A Generalized Family of Exponentiated Composite Distributions

#### 2.2.1. Moments

#### 2.2.2. Survival Function

#### 2.2.3. Hazard Function

#### 2.2.4. Quantile Function

## 3. Special Model: Exponentiated Exponential-Pareto Distribution (EEP)

#### 3.1. Mode

**Proposition**

**1.**

**Proof of Proposition 1.**

#### 3.2. Moments

#### 3.3. Survival Function and Tail Properties

**Proposition**

**2.**

**Proof of Proposition 2.**

#### 3.4. Hazard Function

#### 3.5. Quantile Function

#### 3.6. Median Residual Lifetime

## 4. Parameter Estimation

#### Parameter Search for Two-Parameter EEP Distributions

- Arrange the observations in the sample in increasing order such that ${t}_{(1)}\le {t}_{(2)}\le \dots \le {t}_{(n)}$.
- For $m=1,2,\dots ,n-1$, maximize the objective function $l(\theta ,\eta ;\mathit{t})$ and obtain $({\widehat{\theta}}_{1},{\widehat{\eta}}_{1}),({\widehat{\theta}}_{2},{\widehat{\eta}}_{2}),$ … $,({\widehat{\theta}}_{n-1},{\widehat{\eta}}_{n-1})$ correspondingly.
- Start from $m=1$, check the condition ${t}_{(1)}<{({\widehat{\theta}}_{1})}^{\frac{1}{{\widehat{\eta}}_{1}}}<{t}_{(2)}$. If this is true, then ${\widehat{\theta}}_{1}$ and ${\widehat{\eta}}_{1}$ are the estimates for $\theta $ and $\eta $. If not, go to the next step.
- For $m=2$, check the condition ${t}_{(2)}<{({\widehat{\theta}}_{2})}^{\frac{1}{{\widehat{\eta}}_{2}}}<{t}_{(3)}$. If this is true, then ${\widehat{\theta}}_{2}$ and ${\widehat{\eta}}_{2}$ are the estimates for $\theta $ and $\eta $. Repeat this procedure until the correct m is detected. With the correct m, the corresponding ${\widehat{\theta}}_{m}$ and ${\widehat{\eta}}_{m}$ are the estimates for $\theta $ and $\eta $.

## 5. Goodness-of-Fit Tests with Simulation Studies

- Three different sample sizes: $n=50,100,200$.
- Four different true $\theta $ values: $\theta =0.01,0.1,1,10$.
- Eleven true $\eta $ values: $\eta =0.5,0.6,0.7,0.8,0.9,1,$$1.1,1.2,1.3,1.4,1.5$.

## 6. Application to Real Data

#### 6.1. Second Reactor Pump Data

#### 6.2. Electrical Breakdown of An Insulating Fluid

## 7. Concluding Remark and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

UBT | Upside-down bathtub-shaped |

GEC | Generalized exponentiated composite distributions |

Probability density function | |

CDF | Cumulative distribution function |

EP | Exponential-Pareto distribution |

EEP | Exponentiated exponential-Pareto distribution |

LRT | Likelihood ratio test |

MLE | Maximum likelihood estimates |

AIC | Akaike information criterion |

BIC | Bayesian information criterion |

AICc | Corrected Akaike information criterion |

cAIC | Consistent Akaike information criterion |

KS | Kolmogorov–Smirnoff |

## References

- Bennett, S. Log-Logistic Regression Models for Survival Data. J. R. Stat. Soc. Ser. C (Appl. Stat.)
**1983**, 32, 165–171. [Google Scholar] [CrossRef] - Prentice, R.L. Exponential Survivals with Censoring and Explanatory Variables. Biometrika
**1973**, 60, 279–288. [Google Scholar] [CrossRef] - Prentice, R.L. Linear Rank Tests with Right Censored Data. Biometrika
**1978**, 65, 167–179. [Google Scholar] [CrossRef] - Efron, B. Logistic Regression, Survival Analysis, and the Kaplan-Meier Curve. J. Am. Stat. Assoc.
**1988**, 83, 414–425. [Google Scholar] [CrossRef] - Langlands, A.O.; Pocock, S.J.; Kerr, G.R.; Gore, S.M. Long-term survival of patients with breast cancer: A study of the curability of the disease. Br. Med. J.
**1979**, 2, 1247–1251. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sharma, V.K.; Singh, S.K.; Singh, U. A new upside-down bathtub shaped hazard rate model for survival data analysis. Appl. Math. Comput.
**2014**, 239, 242–253. [Google Scholar] [CrossRef] - de Gusmão, F.R.S.; Ortega, E.M.M.; Cordeiro, G.M. The generalized inverse Weibull distribution. Stat. Pap.
**2011**, 52, 591–619. [Google Scholar] [CrossRef] - Khan, M.S.; King, R. A New Class of Transmuted Inverse Weibull Distribution for Reliability Analysis. Am. J. Math. Manag. Sci.
**2014**, 33, 261–286. [Google Scholar] [CrossRef] - Domma, F.; Condino, F.; Popović, B.V. A new generalized weighted Weibull distribution with decreasing, increasing, upside-down bathtub, N-shape and M-shape hazard rate. J. Appl. Stat.
**2017**, 44, 2978–2993. [Google Scholar] [CrossRef] - Sharma, V.K.; Singh, S.K.; Singh, U.; Agiwal, V. The inverse Lindley distribution: A stress-strength reliability model with application to head and neck cancer data. J. Ind. Prod. Eng.
**2015**, 32, 162–173. [Google Scholar] [CrossRef] - Sharma, V.K.; Singh, S.K.; Singh, U.; Merovci, F. The generalized inverse Lindley distribution: A new inverse statistical model for the study of upside-down bathtub data. Commun. Stat. Theory Methods
**2016**, 45, 5709–5729. [Google Scholar] [CrossRef] - Maurya, S.; Singh, S.; Singh, U. A New Right-Skewed Upside Down Bathtub Shaped Heavy-tailed Distribution and its Applications. J. Mod. Appl. Stat. Methods
**2021**, 19, eP2888. [Google Scholar] [CrossRef] - Liu, B.; Ananda, M.M.A. Analyzing insurance data with an exponentiated composite Inverse-Gamma Pareto Model. Commun. Stat. Theory Methods
**2022**. [Google Scholar] [CrossRef] - Liu, B.; Ananda, M.M.A. A Generalized Family of Exponentiated Composite Distributions. Mathematics
**2022**, 10, 1895. [Google Scholar] [CrossRef] - Scollnik, D.P.M. On composite lognormal-Pareto models. Scand. Actuar. J.
**2007**, 2007, 20–33. [Google Scholar] [CrossRef] - Cooray, K.; Ananda, M.M.A. Modeling actuarial data with a composite lognormal-Pareto model. Scand. Actuar. J.
**2005**. [Google Scholar] [CrossRef] - Teodorescu, S.; Vernic, R. A composite Exponential-Pareto distribution. Ann. “Ovidius” Univ. Constanta Math. Ser.
**2006**, XIV, 99–108. [Google Scholar] - Teodorescu, S.; Vernic, R. Some Composite ExponentialPareto Models for Actuarial Prediction. J. Econ. Forecast.
**2009**, 12, 82–100. [Google Scholar] - Preda, V.; Ciumara, R. On Composite Models: Weibull-Pareto and Lognormal-Pareto. A comparative study. J. Econ. Forecast.
**2006**, 3, 32–46. [Google Scholar] - Cooray, K. The Weibull–Pareto Composite Family with Applications to the Analysis of Unimodal Failure Rate Data. Commun. Stat. Theory Methods
**2009**, 38, 1901–1915. [Google Scholar] [CrossRef] - Deng, M.; Aminzadeh, M.S. Bayesian predictive analysis for Weibull-Pareto composite model with an application to insurance data. Commun. Stat. Simul. Comput.
**2022**, 51, 2683–2709. [Google Scholar] [CrossRef] - Aminzadeh, M.S.; Deng, M. Bayesian predictive modeling for Inverse Gamma-Pareto composite distribution. Commun. Stat. Theory Methods
**2019**, 48, 1938–1954. [Google Scholar] [CrossRef] - Grün, B.; Miljkovic, T. Extending composite loss models using a general framework of advanced computational tools. Scand. Actuar. J.
**2019**, 2019, 642–660. [Google Scholar] [CrossRef] - Nadarajah, S. Exponentiated Pareto distributions. Statistics
**2005**, 39, 255–260. [Google Scholar] [CrossRef] - Sablica, L.; Hornik, K. mistr: A Computational Framework for Mixture and Composite Distributions. R J.
**2020**, 12, 283–299. [Google Scholar] [CrossRef] - Shrahili, M.; Kayid, M. Modeling extreme value data with an upside down bathtub-shaped failure rate model. Open Phys.
**2022**, 20, 484–492. [Google Scholar] [CrossRef] - Burnham, K.P.; Anderson, D.R. Model Selection and Multimodel Inference, 2nd ed.; Springer: New York, NY, USA, 2002. [Google Scholar]
- Bozdogan, H. Mixture-Model Cluster Analysis Using Model Selection Criteria and a New Informational Measure of Complexity. In Proceedings of the First US/Japan Conference on the Frontiers of Statistical Modeling: An Informational Approach: Volume 2 Multivariate Statistical Modeling; Bozdogan, H., Sclove, S.L., Gupta, A.K., Haughton, D., Kitagawa, G., Ozaki, T., Tanabe, K., Eds.; Springer: Dordrecht, The Netherlands, 1994; pp. 69–113. [Google Scholar] [CrossRef]
- Hurvich, C.M.; Tsai, C.L. Regression and time series model selection in small samples. Biometrika
**1989**, 76, 297–307. [Google Scholar] [CrossRef] - Bozdogan, H. Model selection and Akaike’s Information Criterion (AIC): The general theory and its analytical extensions. Psychometrika
**1987**, 52, 345–370. [Google Scholar] [CrossRef] - Suprawhardana, M.S.; Sangadji, P. Total Time on Test Plot Analysis for Mechanical Components of the Rsg-Gas Reactor. Atom Indones
**1999**, 25, 81–90. [Google Scholar] - Meeker, W.Q.; Escobar, L.A.; Pascual, F.G. Statistical Methods for Reliability Data; John Wiley & Sons: Hoboken, NJ, USA, 2022. [Google Scholar]

**Figure 6.**Power comparison of the Likelihood ratio test and asymptotic Wald’s test (the red line represents the asymptotic Wald’s test, and the black line represents the LRT).

**Figure 11.**The fitted hazard function of EEP distribution for the time to electric breakdown data set.

$\mathit{\theta}$ | Sample Size (n) | LRT | Wald’s Test | ||
---|---|---|---|---|---|

$\mathbf{\alpha}=\mathbf{0}.\mathbf{01}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{05}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{01}$ | $\mathbf{\alpha}=\mathbf{0}.\mathbf{05}$ | ||

0.01 | $n=50$ | 0.0110 | 0.0536 | 0.0107 | 0.0484 |

$n=100$ | 0.0125 | 0.0496 | 0.0105 | 0.0477 | |

$n=200$ | 0.0095 | 0.0499 | 0.0101 | 0.0503 | |

0.1 | $n=50$ | 0.0094 | 0.0514 | 0.0087 | 0.0443 |

$n=100$ | 0.0108 | 0.0529 | 0.0104 | 0.0519 | |

$n=200$ | 0.0112 | 0.0495 | 0.0113 | 0.0517 | |

1 | $n=50$ | 0.0116 | 0.0537 | 0.0111 | 0.0469 |

$n=100$ | 0.0120 | 0.0542 | 0.0121 | 0.0523 | |

$n=200$ | 0.0100 | 0.0484 | 0.0113 | 0.0508 | |

10 | $n=50$ | 0.0120 | 0.0544 | 0.0095 | 0.0490 |

$n=100$ | 0.0104 | 0.0523 | 0.0103 | 0.0495 | |

$n=200$ | 0.0121 | 0.0518 | 0.0127 | 0.0523 |

2.160, 0.150, 4.082, 0.746, 0.358, 0.199, 0.402, 0.101, 0.605, 0.954, |

1.359, 0.273, 0.491, 3.465, 0.070, 6.560, 1.060, 0.062, 4.992, 0.614, |

5.320, 0.347, 1.921 |

Model | Estimates | LRT Statistic (p-Value) | Wald Statistic (p-Value) | KS Statistic (p-Value) |
---|---|---|---|---|

EEP | $\theta =0.25516$ $\eta =1.73300$ | 6.32602 (0.01190) | 4.45271 (0.03485) | 0.13043 (0.9924) |

EP | $\theta =0.60707$ | - | - | 0.26087 (0.4218) |

Model | p | AIC | AIC3 | AICc | CAIC |
---|---|---|---|---|---|

EEP | 2 | 72.29528 | 74.29528 | 72.89528 | 76.56627 |

EP | 1 | 76.62130 | 77.62130 | 76.81178 | 78.75679 |

Generalized Heavy-Tailed Pareto | 3 | 69.86288 | 72.86288 | 71.12604 | 76.26936 |

Weighted Weibull–Pareto Composite | 4 | 69.71532 | 73.73368 | 71.95590 | 78.27566 |

7.74, 17.05, 20.46, 21.02 22.66, 43.40, |

47.30, 139.07 144.12, 175.88, 194.90 |

Model | Estimates | LRT Statistic (p-Value) | Wald Statistic (p-Value) | KS Statistic (p-Value) |
---|---|---|---|---|

EEP | $\theta =3326.133$ $\eta =2.473841$ | 4.08993 (0.04314) | 7.090596 (0.008) | 0.54545 (0.0758) |

EP | $\theta =41.39566$ | - | - | 0.90909 (0.0002254) |

Model | p | AIC | AIC3 | AICc | CAIC |
---|---|---|---|---|---|

EEP | 2 | 121.6466 | 123.6466 | 123.1466 | 124.4424 |

EP | 1 | 126.7372 | 127.7372 | 127.1816 | 128.1351 |

Generalized Heavy-Tailed Pareto | 3 | 123.4848 | 125.2912 | 125.7197 | 126.4848 |

Weighted Weibull–Pareto Composite | 4 | 125.2129 | 129.2129 | 131.8795 | 130.8045 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, B.; Ananda, M.M.A.
A New Insight into Reliability Data Modeling with an Exponentiated Composite Exponential-Pareto Model. *Appl. Sci.* **2023**, *13*, 645.
https://doi.org/10.3390/app13010645

**AMA Style**

Liu B, Ananda MMA.
A New Insight into Reliability Data Modeling with an Exponentiated Composite Exponential-Pareto Model. *Applied Sciences*. 2023; 13(1):645.
https://doi.org/10.3390/app13010645

**Chicago/Turabian Style**

Liu, Bowen, and Malwane M. A. Ananda.
2023. "A New Insight into Reliability Data Modeling with an Exponentiated Composite Exponential-Pareto Model" *Applied Sciences* 13, no. 1: 645.
https://doi.org/10.3390/app13010645