A generalized modified Weibull distribution for lifetime modeling.

*(English)*Zbl 1231.62015Summary: A four parameter generalization of the Weibull distribution capable of modeling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates, which are quite common in life time problems and reliability. The new distribution has a number of well-known life time special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull distributions, among others. We derive two infinite sum representations for its moments. The density of the order statistics is obtained. The method of maximum likelihood is used for estimating the model parameters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the proposed distribution.

##### MSC:

62E10 | Characterization and structure theory of statistical distributions |

62N05 | Reliability and life testing |

62F10 | Point estimation |

65C60 | Computational problems in statistics (MSC2010) |

##### Software:

Ox
PDF
BibTeX
XML
Cite

\textit{J. M. F. Carrasco} et al., Comput. Stat. Data Anal. 53, No. 2, 450--462 (2008; Zbl 1231.62015)

Full Text:
DOI

##### References:

[1] | Aarset, M.V., How to identify bathtub hazard rate, IEEE transactions on reliability, 36, 106-108, (1987) · Zbl 0625.62092 |

[2] | Barlow, R.E.; Campo, R., Total time on test processes and applications to failure data analysis, (), 451-481 |

[3] | Bebbington, M.; Lai, C.D.; Zitikis, R., A flexible Weibull extension, Reliability engineering and system safety, 92, 719-726, (2007) |

[4] | Choudhury, A., A simple derivation of moments of the exponentiated Weibull distribution, Metrika, 62, 17-22, (2005) · Zbl 1079.62023 |

[5] | Doornik, J., Ox: an object-oriented matrix programming language, (2007), International Thomson Business Press |

[6] | Glaser, R.E., Bathtub and related failure rate characterizations, Journal of the American statistical association, 75, 667-672, (1980) · Zbl 0497.62017 |

[7] | Gupta, R.D.; Kundu, D., Generalized exponential distributions, Australian and New Zealand journal of statistics, 41, 173-188, (1999) · Zbl 1007.62503 |

[8] | Gupta, R.D.; Kundu, D., Exponentiated exponential distribution: an alternative to gamma and Weibull distributions, Biometrical journal, 43, 117-130, (2001) · Zbl 0997.62076 |

[9] | Haupt, E.; Schabe, H., A new model for a lifetime distribution with bathtub shaped failure rate, Microelectronics and reliability, 32, 633-639, (1992) |

[10] | Hjorth, U., A reliability distribution with increasing, decreasing, constant and bathtub failure rates, Technometrics, 22, 99-107, (1980) · Zbl 0429.62069 |

[11] | Kundu, D.; Rakab, M.Z., Generalized Rayleigh distribution: different methods of estimation, Computational statistics and data analysis, 49, 187-200, (2005) · Zbl 1429.62449 |

[12] | Lai, C.D., Moore, T., Xie, M., 1998. The beta integrated model. In: Proceedings International Workshop on Reliability Modeling and Analysis-From Theory to Practice, pp. 153-159 |

[13] | Lai, C.D.; Xie, M.; Murthy, D.N.P., A modified Weibull distribution, IEEE transactions on reliability, 52, 33-37, (2003) |

[14] | Louzada-Neto, F., Mazucheli, J., Achcar, J.A., 2001. Uma Introdução à Análise de Sobrevivência e Confiabilidade. XXVIII Jornadas Nacionales de Estadística, Chile |

[15] | Mudholkar, G.S.; Srivastava, D.K.; Friemer, M., The exponentiated Weibull family: A reanalysis of the bus-motor-failure data, Technometrics, 37, 436-445, (1995) · Zbl 0900.62531 |

[16] | Mudholkar, G.S.; Srivastava, D.K.; Kollia, G.D., A generalization of the Weibull distribution with application to the analysis of survival data, Journal of the American statistical association, 91, 1575-1583, (1996) · Zbl 0881.62017 |

[17] | Nadarajah, S., On the moments of the modified Weibull distribution, Reliability engineering and system safety, 90, 114-117, (2005) |

[18] | Nadarajah, S.; Kotz, S., On some recent modifications of Weibull distribution, IEEE transactions reliability, 54, 561-562, (2005) |

[19] | Nelson, W., Lifetime data analysis, (1982), Wiley New York |

[20] | Perdoná, G.S.C., 2006. Modelos de Riscos Aplicados à Análise de Sobrevivência. Doctoral Thesis, Institute of Computer Science and Mathematics, University of São Paulo, Brasil (in Portuguese) |

[21] | Pham, H.; Lai, C.D., On recent generalizations of the Weibull distribution, IEEE transactions on reliability, 56, 454-458, (2007) |

[22] | Rajarshi, S.; Rajarshi, M.B., Bathtub distributions: A review, Communications in statistics-theory and methods, 17, 2521-2597, (1988) · Zbl 0696.62027 |

[23] | Silva, A.N.F., 2004. Estudo evolutivo das crianças expostas ao HIV e notificadas pelo núcleo de vigilância epidemiológica do HCFMRP-USP. M.Sc. Thesis. University of São Paulo, Brazil |

[24] | Xie, M.; Lai, C.D., Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function, Reliability engineering and system safety, 52, 87-93, (1995) |

[25] | Xie, M.; Tang, Y.; Goh, T.N., A modified Weibull extension with bathtub failure rate function, Reliability engineering and system safety, 76, 279-285, (2002) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.