A Resilience Parameter Model Generated by a Compound Distribution
Department of Statistics, University of Isfahan
Isfahan, 81746-73441, Iran
E-mail: h.bidram@sci.ui.ac.ir
Received:
Received: 19 July 2013; revised: 15 November 2013; accepted: 18 November 2013; published online: 9 December 2013
DOI: 10.12921/cmst.2013.19.04.217-228
OAI: oai:lib.psnc.pl:458
Abstract:
In this paper, we shall attempt to extend the generalized exponential geometric distribution of Silva et al. [1]. The new four-parameter distribution also generalizes the Weibull-geometric distribution of Barreto-Souza et al. [2],
exponentiated Weibull, and several other lifetime distributions as special cases. A useful characteristic of the new distribution is that its failure rate function can have different shapes. We first study certain basic distributional properties of the new distribution and provide closed form expressions for its moment generating function and moments. General expressions are also obtained for the order statistics densities and stress-strength parameter. Our findings happen to enfold several known results as special cases. The model parameters are estimated by the maximum likelihood method and the Fisher information matrix is discussed. Finally, the model is applied to a real data set and its advantage over some rival models is illustrated.
Key words:
Beta Weibull, Beta Weibull-geometric, Maximum likelihood estimation, Order statistics, Resilience parameter family, Stress-strength parameter
References:
[1] R.B. Silva, W. Barreto-Souza, and G.M. Cordeiro, A new
distribution with decreasing, increasing, and upside-down
bathtub failure rate, Comput. Statist. Data Anal. 54, 935-944
(2010).
[2] W. Barreto-Souza, A.L. de Morais, and G.M. Cordeiro, The
Weibull-geometric distribution, J. Stat. Comput. Simul. 81,
645-657 (2011).
[3] A.W. Marshall and I. Olkin, Life distributions: Structure
of nonparametric, semiparametric, and parametric families,
Springer Science+Business Media, LLC, New York, 2007.
[4] G.S. Mudholkar and D.K. Srivastava, Exponentiated Weibull
family for analyzing bathtub failure -rate data, Trans. Reliab.
42, 299-302 (1993).
[5] G.S. Mudholkar, D.K. Srivastava, and M. Freimer, The expo-
nentiated Weibull family: A reanalysis of the bus-motor-failure
data, Technometrics 37, 436-445 (1995).
[6] G.S. Mudholkar and A.D. Hutson, The exponentiated Weibull
family: Some properties and a flood data application, Comm.
Statist. Theory Methods 25, 3059-3083 (1996).
[7] R.D. Gupta and D. Kundu, Generalized exponential distribu-
tions, Aust. N. Z. J. Stat. 41, 173-188 (1999).
[8] S. Nadarajah and S. Kotz, The exponentiated type distribu-
tions, Acta Appl. Math. 92, 97-111 (2006).
[9] W. Barreto-Souza and F. Cribari-Neto, A generalization of
the exponential-Poisson distribution, Statist. Probab. Lett. 79,
2493-2500 (2009).
[10] J.M.F. Carrasco, E.M.M. Ortega, and G.M. Cordeiro, A gen-
eralized modified Weibull distribution for lifetime modeling,
Comput. Statist. Data Anal. 53, 450-462 (2008).
[11] C. Kus, A new lifetime distribution, Comput. Statist. Data
Anal. 51, 4497-4509 (2007).
[12] C.D. Lai, M. Xie, and D.N.P. Murthy, A modified Weibull
distribution, Trans. Reliab. 52, 33-37 (2003).
[13] K. Adamidis and S. Loukas, A lifetime distribution with de-
creasing failure rate, Statist. Probab. Lett. 39, 35-42 (1998).
[14] H. Bidram, J. Behboodian, and M. Towhidi, A new general-
ized exponential geometric distribution, Comm. Statist. The-
ory Methods. 42, 528-542 (2013).
[15] A. W. Marshall and I. Olkin, A new method for adding a pa-
rameter to a family of distributions with application to the
exponential and Weibull families, Biometrika 84, 641-652
(1997).
[16] V.K. Rohatgi, Distribution of order statistics with random
sample size. Comm. Statist. Theory Methods 16, 3739-3743
(1987).
[17] S. Dharmadhikary and K. Joag-dev, Unimodality, convexity,
and applications, Academic Press, Boston, 1998.
[18] R.E. Glaser, R.E., Bathtub and related failure rate character-
izations, J. Am. Stat. Assoc. 75, 667-672 (1980).
[19] G.M. Cordeiro, A.B. Simas, and B.D. Stosic, Closed form ex-
pressions for moments of the beta Weibull distribution. Annals
of the Brazilian Academy of Sciences. 83, 357-373 (2011).
[20] F. Famoye, C. Lee, and O. Olumolade, The beta-Weibull
distribution, J. Statist. Theory Appl. 4, 121-136 (2005).
[21] C. Lee, F. Famoye, and O. Olumolade, Beta-Weibull distri-
bution: Some properties and applications to censored data, J.
Mod. Appl. Statist. Methods 6, 173-186 (2007).
[22] H. Bidram, J. Behboodian, and M. Towhidi, The beta Weibull-
geometric distribution, J. Statist. Comput. Simul. 83, 52-67
(2013).
[23] A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi,
Higher Transcendental Functions, McGraw-Hill, New York,
1953.
[24] T. S. Ferguson, A course in large sample theory, Chapman
and Hall, London, 1996.
[25] R.L. Smith and J.C. Naylor, A comparison of maximum likeli-
hood and Bayesian estimators for the three-parameter Weibull
distribution, J. Appl. Stat. 36, 358-369 (1987).
[26] W. Barreto-Souza, A.H. S. Santos, and G.M. Cordeiro,
The beta generalized exponential distribution, J. Statist. Com-
put. Simul. 80, 159-172 (2010).
In this paper, we shall attempt to extend the generalized exponential geometric distribution of Silva et al. [1]. The new four-parameter distribution also generalizes the Weibull-geometric distribution of Barreto-Souza et al. [2],
exponentiated Weibull, and several other lifetime distributions as special cases. A useful characteristic of the new distribution is that its failure rate function can have different shapes. We first study certain basic distributional properties of the new distribution and provide closed form expressions for its moment generating function and moments. General expressions are also obtained for the order statistics densities and stress-strength parameter. Our findings happen to enfold several known results as special cases. The model parameters are estimated by the maximum likelihood method and the Fisher information matrix is discussed. Finally, the model is applied to a real data set and its advantage over some rival models is illustrated.
Key words:
Beta Weibull, Beta Weibull-geometric, Maximum likelihood estimation, Order statistics, Resilience parameter family, Stress-strength parameter
References:
[1] R.B. Silva, W. Barreto-Souza, and G.M. Cordeiro, A new
distribution with decreasing, increasing, and upside-down
bathtub failure rate, Comput. Statist. Data Anal. 54, 935-944
(2010).
[2] W. Barreto-Souza, A.L. de Morais, and G.M. Cordeiro, The
Weibull-geometric distribution, J. Stat. Comput. Simul. 81,
645-657 (2011).
[3] A.W. Marshall and I. Olkin, Life distributions: Structure
of nonparametric, semiparametric, and parametric families,
Springer Science+Business Media, LLC, New York, 2007.
[4] G.S. Mudholkar and D.K. Srivastava, Exponentiated Weibull
family for analyzing bathtub failure -rate data, Trans. Reliab.
42, 299-302 (1993).
[5] G.S. Mudholkar, D.K. Srivastava, and M. Freimer, The expo-
nentiated Weibull family: A reanalysis of the bus-motor-failure
data, Technometrics 37, 436-445 (1995).
[6] G.S. Mudholkar and A.D. Hutson, The exponentiated Weibull
family: Some properties and a flood data application, Comm.
Statist. Theory Methods 25, 3059-3083 (1996).
[7] R.D. Gupta and D. Kundu, Generalized exponential distribu-
tions, Aust. N. Z. J. Stat. 41, 173-188 (1999).
[8] S. Nadarajah and S. Kotz, The exponentiated type distribu-
tions, Acta Appl. Math. 92, 97-111 (2006).
[9] W. Barreto-Souza and F. Cribari-Neto, A generalization of
the exponential-Poisson distribution, Statist. Probab. Lett. 79,
2493-2500 (2009).
[10] J.M.F. Carrasco, E.M.M. Ortega, and G.M. Cordeiro, A gen-
eralized modified Weibull distribution for lifetime modeling,
Comput. Statist. Data Anal. 53, 450-462 (2008).
[11] C. Kus, A new lifetime distribution, Comput. Statist. Data
Anal. 51, 4497-4509 (2007).
[12] C.D. Lai, M. Xie, and D.N.P. Murthy, A modified Weibull
distribution, Trans. Reliab. 52, 33-37 (2003).
[13] K. Adamidis and S. Loukas, A lifetime distribution with de-
creasing failure rate, Statist. Probab. Lett. 39, 35-42 (1998).
[14] H. Bidram, J. Behboodian, and M. Towhidi, A new general-
ized exponential geometric distribution, Comm. Statist. The-
ory Methods. 42, 528-542 (2013).
[15] A. W. Marshall and I. Olkin, A new method for adding a pa-
rameter to a family of distributions with application to the
exponential and Weibull families, Biometrika 84, 641-652
(1997).
[16] V.K. Rohatgi, Distribution of order statistics with random
sample size. Comm. Statist. Theory Methods 16, 3739-3743
(1987).
[17] S. Dharmadhikary and K. Joag-dev, Unimodality, convexity,
and applications, Academic Press, Boston, 1998.
[18] R.E. Glaser, R.E., Bathtub and related failure rate character-
izations, J. Am. Stat. Assoc. 75, 667-672 (1980).
[19] G.M. Cordeiro, A.B. Simas, and B.D. Stosic, Closed form ex-
pressions for moments of the beta Weibull distribution. Annals
of the Brazilian Academy of Sciences. 83, 357-373 (2011).
[20] F. Famoye, C. Lee, and O. Olumolade, The beta-Weibull
distribution, J. Statist. Theory Appl. 4, 121-136 (2005).
[21] C. Lee, F. Famoye, and O. Olumolade, Beta-Weibull distri-
bution: Some properties and applications to censored data, J.
Mod. Appl. Statist. Methods 6, 173-186 (2007).
[22] H. Bidram, J. Behboodian, and M. Towhidi, The beta Weibull-
geometric distribution, J. Statist. Comput. Simul. 83, 52-67
(2013).
[23] A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi,
Higher Transcendental Functions, McGraw-Hill, New York,
1953.
[24] T. S. Ferguson, A course in large sample theory, Chapman
and Hall, London, 1996.
[25] R.L. Smith and J.C. Naylor, A comparison of maximum likeli-
hood and Bayesian estimators for the three-parameter Weibull
distribution, J. Appl. Stat. 36, 358-369 (1987).
[26] W. Barreto-Souza, A.H. S. Santos, and G.M. Cordeiro,
The beta generalized exponential distribution, J. Statist. Com-
put. Simul. 80, 159-172 (2010).
