×

zbMATH — the first resource for mathematics

The generalized inverse Weibull distribution. (English) Zbl 1230.62014
Summary: The inverse Weibull distribution has the ability to model failure rates which are quite common in reliability and biological studies. A three-parameter generalized inverse Weibull distribution with decreasing and unimodal failure rate is introduced and studied. We provide a comprehensive treatment of the mathematical properties of the new distribution including expressions for the moment generating function and the \(r\) th generalized moment. The mixture model of two generalized inverse Weibull distributions is investigated. The identifiability property of the mixture model is demonstrated. For the first time, we propose a location-scale regression model based on the log-generalized inverse Weibull distribution for modeling life time data. In addition, we develop some diagnostic tools for sensitivity analysis. Two applications of real data are given to illustrate the potentiality of the proposed regression model.

MSC:
62E10 Characterization and structure theory of statistical distributions
62N05 Reliability and life testing
62J20 Diagnostics, and linear inference and regression
62F10 Point estimation
62G30 Order statistics; empirical distribution functions
62H12 Estimation in multivariate analysis
65C60 Computational problems in statistics (MSC2010)
Software:
Ox
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aarset MV (1987) How to identify bathtub hazard rate. IEEE Trans Reliab 36: 106–108 · Zbl 0625.62092 · doi:10.1109/TR.1987.5222310
[2] AL-Hussaini EK, Sultan KS (2001) Reliability and hazard based on finite mixture models. In: Balakrishnan N, Rao CR (eds) Handbook of statistics, vol 20. Elsevier, Amsterdam, pp 139–183
[3] Barakat HM, Abdelkader YH (2004) Computing the moments of order statistics from nonidentical random variables. Stat Methods Appl 13: 15–26 · Zbl 1056.62012
[4] Barlow WE, Prentice RL (1988) Residuals for relative risk regression. Biometrika 75: 65–74 · Zbl 0632.62102 · doi:10.1093/biomet/75.1.65
[5] Barreto ML, Santos LMP, Assis AMO, Araújo MPN, Farenzena GG, Santos PAB, Fiaccone RL (1994) Effect of vitamin A supplementation on diarrhoea and acute lower-respiratory-tract infections in young children in Brazil. Lancet 344: 228–231 · doi:10.1016/S0140-6736(94)92998-X
[6] Chandra S (1977) On the mixtures of probability distributions. Scand J Stat 4: 105–112 · Zbl 0369.60024
[7] Cook RD (1986) Assesment of local influence (with discussion). J R Stat Soc B 48: 133–169 · Zbl 0608.62041
[8] Doornik J (2007) Ox: an object-oriented matrix programming language. International Thomson Bussiness Press, London
[9] Drapella A (1993) Complementary Weibull distribution: unknown or just forgotten. Qual Reliab Eng Int 9: 383–385 · doi:10.1002/qre.4680090426
[10] Everitt BS, Hand DJ (1981) Finite mixture distributions. Chapman and Hall, London · Zbl 0466.62018
[11] Fleming TR, Harrington DP (1991) Counting process and survival analysis. Wiley, New York
[12] Ghosh SK, Ghosal S (2006) Semiparametric accelerated failure time models for censored data. In: Upadhyay SK, Singh U, Dey DK (eds) Bayesian statistics and its applications. Anamaya Publishers, New Delhi, pp 213–229
[13] Hosmer DW, Lemeshow S (1999) Applied survival analysis. Wiley, New York · Zbl 0966.62071
[14] Jiang R, Zuo MJ, Li HX (1999) Weibull and Weibull inverse mixture models allowing negative weights. Reliab Eng Syst Saf 66: 227–234 · doi:10.1016/S0951-8320(99)00037-X
[15] Jiang R, Murthy DNP, Ji P (2001) Models involving two inverse Weibull distributions. Reliab Eng Syst Saf 73: 73–81 · doi:10.1016/S0951-8320(01)00030-8
[16] Jin Z, Lin DY, Wei LJ, Ying Z (2003) Rank-based inference for the accelerated failure time model. Biometrika 90: 341–353 · Zbl 1034.62103 · doi:10.1093/biomet/90.2.341
[17] Keller AZ, Kamath AR (1982) Reliability analysis of CNC machine tools. Reliab Eng 3: 449–473 · doi:10.1016/0143-8174(82)90036-1
[18] Krall J, Uthoff V, Harley J (1975) A step-up procedure for selecting variables associated with survival. Reliab Eng Syst Saf 73: 73–81 · Zbl 0308.62098
[19] Lawless JF (2003) Statistical models and methods for lifetime data. Wiley, New York · Zbl 1015.62093
[20] Lesaffre E, Verbeke G (1998) Local influence in linear mixed models. Biometrics 54: 570–582 · Zbl 1058.62623 · doi:10.2307/3109764
[21] Maclachlan GJ, Krishnan T (1997) The EM algorithm and extensions. Wiley, New York
[22] Maclachlan G, Peel D (2000) Finite mixture models. Wiley, New York · Zbl 0963.62061
[23] Mudholkar GS, Kollia GD (1994) Generalized Weibull family: a structural analysis. Commun Stat Ser A 23: 1149–1171 · Zbl 0825.62132 · doi:10.1080/03610929408831309
[24] Mudholkar GS, Srivastava DK, Kollia GD (1996) A generalization of the Weibull distribution with application to the analysis of survival data. J Am Stat Assoc 91: 1575–1583 · Zbl 0881.62017 · doi:10.1080/01621459.1996.10476725
[25] Nadarajah S (2006) The exponentiated Gumbel distribution with climate application. Environmetrics 17: 13–23 · doi:10.1002/env.739
[26] Ortega EMM, Paula GA, Bolfarine H (2008) Deviance residuals in generalized log-gamma regression models with censored observations. J Stat Comput Simul 78: 747–764 · Zbl 1145.62081 · doi:10.1080/00949650701282465
[27] Prudnikov AP, Brychkov YA, Marichev OI (1986) Integrals and series. Gordon and Breach Science Publishers, New York
[28] Sultan KS, Ismail MA, Al-Moisheer AS (2007) Mixture of two inverse Weibull distributions: properties and estimation. Comput Stat Data Anal 51: 5377–5387 · Zbl 1445.62027 · doi:10.1016/j.csda.2006.09.016
[29] Therneau TM, Grambsch PM, Fleming TR (1990) Martingale-based residuals for survival models. Biometrika 77: 147–160 · Zbl 0692.62082 · doi:10.1093/biomet/77.1.147
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.