Abstract
In this paper, the identifiability of a finite mixture of generalized exponential distributions (GE(τ, α)) is proved and the maximum likelihood estimates (MLE’s) of the parameters are obtained using EM algorithm based on a general form of right-censored failure times. The results are specialized to type-I and type-II censored samples. A real data set is introduced and analyzed using a mixture of two GE(τ, α) distributions and also using a mixture of two Weibull(α, β) distributions. A comparison is carried out between the mentioned mixtures based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the GE(τ, α) mixture model fits the data better than the other mixture model.
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References
Ahmad KE (1988) Identifiability of finite mixtures using a new transform. Ann Inst Stat Math 40(2): 261–265
Ahmad KE, AL-Hussaini EK (1982) Remarks on the nonidentifiability of mixtures of distributions. Ann Inst Stat Math 34(A): 543–544
Ahmad Mahir R, Ali SA (2009) Combining two Weibull distributions using a mixing parameter. Eur J Sci Res 31(2): 296–305
Ateya SF (2011) Estimation under finite mixture of truncated type I generalized logistic components model based on censored data via EM algorithm. Int Math Forum 6(67): 3323–3341
Chandra S (1977) On mixture of probability distributions. Scand J Stat 4: 105–112
Chen DG, Lio YL (2010) Parameter estimations for generalized exponen-tial distribution under progressive type-I interval censoring. Comput Stat Data Anal 54: 1581–1591
Dempster AR, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via EM algorithm. J R Stat Soc B 39: 1–38
Günter M (1963) Three essays in econometrics lectures held in bombay and calcutta in September/October 1962. Stat Papers 4(1): 1–37
Gupta RD, Kundu D (1999) Generalized exponential distribution. Aust NZ J Stat 41(2): 173–188
Gupta RD, Kundu D (2001) Generalized exponential distribution. Different method of estimations. J Stat Comput Simul 69: 315–338
Gupta RD, Kundu D (2001) Generalized exponential distribution. An alternative to gamma or Weibull distribution. Biomet J 43: 117–130
Gupta RD, Kundu D (2002) Generalized exponential distribution. J Appl Stat Soc 1: 101–118
Gupta RD, Kundu D (2003) Closeness between the gamma and generalized exponential distributions. Commun Stat Theory Methods 32: 705–722
Gupta RD, Kundu D (2003) Discriminating between the weibull and generalized exponential distributions. Comput Stat Data Anal 43: 179–196
Gupta RD, Kundu D (2004) Discriminating between the gamma and generalized exponential distributions. J Statist Comput Simul 74: 107–122
Gupta RD, Kundu D (2007) Generalized exponential distribution: existing results and some recent developments. J Stat Plan Infer 137(11): 3537–3547
Hartley H (1958) Maximum likelihood estimation from incomplete data. Biometrics 14: 174–194
Jaheen ZF (2004) Empirical Bayes inference for generalized exponential distribution based on records. Commun Stat Theory Methods 33(8): 1851–1861
Kalbfleisch JD, Prentice RL (1980) The statistical analysis of failure time data, 2nd edn, Wiley, New York
Kim C, Song S (2010) Bayesian estimation of the parameters of the generalized exponential distribution from doubly censored samples. Stat Papers 51(3): 583–597
Kim C, Song S (2011) Bayesian estimation for the exponentiated Weibull model under Type II progressive censoring. Stat Papers 52(1): 53–70
Kundu D, Gupta RD, Manglick A (2005) Discriminating between the log–normal and generalized exponential distributions. J Stat Plan Infer 127: 213–227
Kundu D, Gupta RD (2008) Generalized exponential distribution: Bayesian estimations. Comput Stat Data Anal 52(4): 1873–1883
Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, New York
McLachlan GJ, Krishnan T (1997) The EM algorithm and extensions. Wiley, New York
Minka T (1998) Expectation–Maximization as lower bound maximization. Tuto rial published on the web at http://www-white.media.mit.edu/tpminka/paper-s/em.html.aa
Mohanty NC (1972) On the identifiability of finite mixture of Laguerre distri butions. IEEE Trans Inform Theory 18: 514–515
Mudholkar GS, Hutson AD (1996) The exponentiated weibull family, some properties and a flood data applications. Commun Stat Theory Methods 25: 3059–3083
Mudholkar GS, Srivastava DK (1993) The exponentiated weibull family for analyzing bathtub failure-rate data. IEEE Trans Reliab 42: 299–302
Mudholkar GS, Srivastava DK, Freimer M (1995) The exponentiated weibull family, a reanalysis of the bus-motor failure data. Technometrics 37: 436–445
Neal R, Hinton G (1998) A view of the EM algorithm that justifies incremental, sparse, and other variants. In Jordan M, ed. Learning in graphical models. Kluwer Academic, Norwell
Raqab MZ (2002) Inference for generalized exponential distributions based on record statistics. J Stat Plan Infer 104: 339–350
Raqab MZ, Ahsanullah M (2001) Estimation of the location and scale parameters of the generalized exponential distributions based on order statistics. J Stat Comput Simul 69: 109–123
Raqab MZ, Madi MT (2005) Bayesian inference for the generalized exponential distribution. J Stat Comput Simul 75(10): 841–852
Rennie RR (1972) On the interdependence of the identifiability of multivariate mixtures and the identifiability of the marginal mixtures. Sankhy, Ser A 34: 449–452
Sarhan AM (2007) Analysis of incomplete, censored data in competing risks models with generalized exponential distributions. IEEE Trans Reliab 56: 132–138
Singh R, Singh SK, Singh U, Singh GP (2008) Bayes estimator of generalized exponential parameters under LINEX loss function using Lindley’s approximation. Data Sci J 7: 65–75
Swagata Nandi, Isha Dewan (2010) An EM algorithm for estimating the parameters of bivariate Weibull distribution under random censoring. Comput Stat Data Anal 54: 1559–1569
Tanner M (1996) Tools for statistical inference, 3rd edn. Springer, New York
Teicher H (1961) Identifiability of mixtures. Ann Math Stat 33: 244–248
Teicher H (1963) Identifiability of finite mixtures. Ann Math Stat 34: 1265–1269
Teicher H (1967) Identifiability of product measures. Ann Math Stat 38: 1300–1302
Titterington DM, Smith AFM, Makov UE (1985) Statistical analysis of finite mixture distributions. Wiley, New York
Yakowitz SJ, Spragins JD (1968) On the identifiability of finite mixtures. Ann Math Stat 39: 209–214
Yarmohammadi M (2010) Classical and Bayesian estimations on the generalized exponential distribution using censored data. Int J Math Anal 4(29): 1417–1431
Zheng G (2002) On the fisher information matrix in type-II censored data from the exponentiated exponential family. Biomet J 44: 353–357
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Ateya, S.F. Maximum likelihood estimation under a finite mixture of generalized exponential distributions based on censored data. Stat Papers 55, 311–325 (2014). https://doi.org/10.1007/s00362-012-0480-z
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DOI: https://doi.org/10.1007/s00362-012-0480-z
Keywords
- Generalized exponential distribution
- Weibull distribution
- Kolmogorov–Smirnov test
- Identifiability of finite mixture distributions
- Generalized right-censored failure times
- Random right-censored failure times
- Type-I and type-II censoring
- EM algorithm