DOI QR코드

DOI QR Code

Inverted exponentiated Weibull distribution with applications to lifetime data

  • Lee, Seunghyung (Department of Statistics, Pusan National University) ;
  • Noh, Yunhwan (Department of Statistics, Pusan National University) ;
  • Chung, Younshik (Department of Statistics, Pusan National University)
  • 투고 : 2016.11.28
  • 심사 : 2017.04.07
  • 발행 : 2017.05.31

초록

In this paper, we introduce the inverted exponentiated Weibull (IEW) distribution which contains exponentiated inverted Weibull distribution, inverse Weibull (IW) distribution, and inverted exponentiated distribution as submodels. The proposed distribution is obtained by the inverse form of the exponentiated Weibull distribution. In particular, we explain that the proposed distribution can be interpreted by Marshall and Olkin's book (Lifetime Distributions: Structure of Non-parametric, Semiparametric, and Parametric Families, 2007, Springer) idea. We derive the cumulative distribution function and hazard function and calculate expression for its moment. The hazard function of the IEW distribution can be decreasing, increasing or bathtub-shaped. The maximum likelihood estimation (MLE) is obtained. Then we show the existence and uniqueness of MLE. We can also obtain the Bayesian estimation by using the Gibbs sampler with the Metropolis-Hastings algorithm. We also give applications with a simulated data set and two real data set to show the flexibility of the IEW distribution. Finally, conclusions are mentioned.

키워드

참고문헌

  1. Birnbaum ZW and Saunders SC (1969). Estimation for a family of life distributions with applications to fatigue, Journal of Applied Probability, 6, 328-347. https://doi.org/10.2307/3212004
  2. Chib S and Greenberg E (1995). Understanding the Metropolis-Hastings algorithm, The American Statistician, 49, 327-335.
  3. De Gusmao FRS, Ortega EMM, and Cordeiro GM (2011). The generalized inverse Weibull distribution, Statistical Papers, 52, 591-619. https://doi.org/10.1007/s00362-009-0271-3
  4. De Gusmao FRS, Ortega EMM, and Cordeiro GM (2012). Reply to the "Letter to the Editor" of M. C. Jones, Statistical Papers, 53, 253-254. https://doi.org/10.1007/s00362-012-0441-6
  5. Elbatal I, Condino F, and Domma F (2016). Reflected generalized beta inverse Weibull distribution: definition and properties, Sankhya B, 78, 316-340. https://doi.org/10.1007/s13571-015-0114-2
  6. Flaih A, Elsalloukh H, Mendi E, and Milanova M (2012). The exponentiated inverted Weibull distribution, Applied Mathematics and Information Sciences, 6, 167-171.
  7. Gelfand AE and Smith AFM (1990). Sampling-based approaches to calculating marginal densities, Journal of the American Statistical Association, 85, 398-409. https://doi.org/10.1080/01621459.1990.10476213
  8. Gupta RD and Kundu D (1999). Theory & methods: generalized exponential distributions, Australian and New Zealand Journal of Statistics, 41, 173-188. https://doi.org/10.1111/1467-842X.00072
  9. Jones MC (2012). Letter to the Editor, Statistical Papers, 53, 251. https://doi.org/10.1007/s00362-012-0440-7
  10. Khan MS, King R, and Hudson IL (2015). Transmuted generalized exponential distribution: A generalisation of the exponential distribution with applications to survival data, Communications in Statistics - Simulation and Computation, Manuscript just-accepted for publication.
  11. Khan MS, King R, and Hudson IL (2017). Transmuted Weibull distribution: properties and estimation, Communications in Statistics - Theory and Methods, 46, 5394-5418. https://doi.org/10.1080/03610926.2015.1100744
  12. Krishna H and Kumar K (2013). Reliability estimation in generalized inverted exponential distribution with progressively type II censored sample, Journal of Statistical Computation and Simulation, 83, 1007-1019. https://doi.org/10.1080/00949655.2011.647027
  13. Lawless JF (1982). Statistical Models and Methods for Lifetime Data, Wiley, New York.
  14. Lee S (2014). Bayesian Estimation of Parameters of Inverted Exponentiated Weibull Distribution (Master's Thesis), Pusan National University, Pusan.
  15. Marshall AW and Olkin I (2007). Lifetime Distributions: Structure of Non-parametric, Semiparametric, and Parametric Families, Springer, New York.
  16. Mudholkar GS and Srivastava DK (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability, 42, 299-302. https://doi.org/10.1109/24.229504
  17. Rastogi MK and Tripathi YM (2014). Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring, Journal of Applied statistics, 41, 2375-2405. https://doi.org/10.1080/02664763.2014.910500
  18. Singh SK, Singh U, and Kumar D (2013). Bayesian estimation of parameters of inverse Weibull distribution, Journal of Applied statistics, 40, 1597-1607. https://doi.org/10.1080/02664763.2013.789492
  19. Singh U, Gupta PK, and Upadhyay SK (2002). Estimation of exponentiated Weibull shape parameters under LINEX loss function, Communications in Statistics - Simulation Computation, 31, 523-537. https://doi.org/10.1081/SAC-120004310
  20. Singh U, Gupta PK, and Upadhyay SK (2005). Estimation of three-parameter exponentiated-Weibull distribution under type-II censoring, Journal of Statistical Planning and Inference, 134, 350-372. https://doi.org/10.1016/j.jspi.2004.04.018