### Abstract

Two and three-parameter Weibull distribution is considered a flexible and useful distribution for adequately representing unimodal frequency distribution of failure times, but sometimes these distributions do not accurately represent the failure times data set. In such cases mixture of two or three Weibull distributions developed here provide very good fits for these mixture distributions. In this paper a mixture of two and three Weibull distributions were used to analyze the data of failure times. The suitability of the distributions is judged from the various tests-of-fit commonly used in the specialized literature on failure times data. The shapes of the density and hazard functions were used in addition to another procedure using goodness of fit tests based on the empirical distribution function are used to find the suitability fits of the data of failure times. These measurements are; coefficient of determination R^{2}, sum of squares due to error SSE, mean square error MSE and root mean square error RMSE. Maximum likelihood estimation MLE was used to estimate the parameters. We found that two- and three-component mixture Weibull distribution provides suitable fits for the failure time data studied based on the shapes of density and hazard functions. A high value of R^{2} and low SSE, MSE and RMSE were obtained for five cases indicating suitable fit. It was concluded that the mixture Weibull distributions provide very flexible models for the proposed failure times data. It was also found that increasing the number of components resulting in increasing the number of parameters can have a negative effect on the values of R^{2}, SSE, MSE and RMSE.

Original language | English |
---|---|

Pages (from-to) | 11358-11364 |

Number of pages | 7 |

Journal | Applied Mathematics and Computation |

Volume | 219 |

Issue number | 24 |

DOIs | |

Publication status | Published - 2013 |

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### Keywords

- Coefficient of determination (R)
- Maximum likelihood estimation
- Mean square error
- Mixture Weibull distribution
- Shape of the density and hazard function

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*Applied Mathematics and Computation*,

*219*(24), 11358-11364. https://doi.org/10.1016/j.amc.2013.05.062

**Mixture Weibull distributions for fitting failure times data.** / Razali, Ahmad Mahir; Al-Wakeel, Ali A.

Research output: Contribution to journal › Article

*Applied Mathematics and Computation*, vol. 219, no. 24, pp. 11358-11364. https://doi.org/10.1016/j.amc.2013.05.062

}

TY - JOUR

T1 - Mixture Weibull distributions for fitting failure times data

AU - Razali, Ahmad Mahir

AU - Al-Wakeel, Ali A.

PY - 2013

Y1 - 2013

N2 - Two and three-parameter Weibull distribution is considered a flexible and useful distribution for adequately representing unimodal frequency distribution of failure times, but sometimes these distributions do not accurately represent the failure times data set. In such cases mixture of two or three Weibull distributions developed here provide very good fits for these mixture distributions. In this paper a mixture of two and three Weibull distributions were used to analyze the data of failure times. The suitability of the distributions is judged from the various tests-of-fit commonly used in the specialized literature on failure times data. The shapes of the density and hazard functions were used in addition to another procedure using goodness of fit tests based on the empirical distribution function are used to find the suitability fits of the data of failure times. These measurements are; coefficient of determination R2, sum of squares due to error SSE, mean square error MSE and root mean square error RMSE. Maximum likelihood estimation MLE was used to estimate the parameters. We found that two- and three-component mixture Weibull distribution provides suitable fits for the failure time data studied based on the shapes of density and hazard functions. A high value of R2 and low SSE, MSE and RMSE were obtained for five cases indicating suitable fit. It was concluded that the mixture Weibull distributions provide very flexible models for the proposed failure times data. It was also found that increasing the number of components resulting in increasing the number of parameters can have a negative effect on the values of R2, SSE, MSE and RMSE.

AB - Two and three-parameter Weibull distribution is considered a flexible and useful distribution for adequately representing unimodal frequency distribution of failure times, but sometimes these distributions do not accurately represent the failure times data set. In such cases mixture of two or three Weibull distributions developed here provide very good fits for these mixture distributions. In this paper a mixture of two and three Weibull distributions were used to analyze the data of failure times. The suitability of the distributions is judged from the various tests-of-fit commonly used in the specialized literature on failure times data. The shapes of the density and hazard functions were used in addition to another procedure using goodness of fit tests based on the empirical distribution function are used to find the suitability fits of the data of failure times. These measurements are; coefficient of determination R2, sum of squares due to error SSE, mean square error MSE and root mean square error RMSE. Maximum likelihood estimation MLE was used to estimate the parameters. We found that two- and three-component mixture Weibull distribution provides suitable fits for the failure time data studied based on the shapes of density and hazard functions. A high value of R2 and low SSE, MSE and RMSE were obtained for five cases indicating suitable fit. It was concluded that the mixture Weibull distributions provide very flexible models for the proposed failure times data. It was also found that increasing the number of components resulting in increasing the number of parameters can have a negative effect on the values of R2, SSE, MSE and RMSE.

KW - Coefficient of determination (R)

KW - Maximum likelihood estimation

KW - Mean square error

KW - Mixture Weibull distribution

KW - Shape of the density and hazard function

UR - http://www.scopus.com/inward/record.url?scp=84879596703&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879596703&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2013.05.062

DO - 10.1016/j.amc.2013.05.062

M3 - Article

VL - 219

SP - 11358

EP - 11364

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

IS - 24

ER -