×

zbMATH — the first resource for mathematics

A generalization of the exponential-Poisson distribution. (English) Zbl 1176.62005
Summary: The two-parameter distribution known as exponential-Poisson (EP) distribution, which has a decreasing failure rate, was introduced by C. Kus [Comput. Stat. Data Anal. 51, No. 9, 4497–4509 (2007; Zbl 1162.62309)]. We generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the \(i\) th order statistic.
We derive the \(r\) th raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher’s information matrix. Furthermore, expressions for the Rényi and Shannon entropies are given and an application using a real data set is presented. Finally, simulation results on maximum likelihood estimation are presented.

MSC:
62E10 Characterization and structure theory of statistical distributions
62N05 Reliability and life testing
62F10 Point estimation
62G30 Order statistics; empirical distribution functions
62N02 Estimation in survival analysis and censored data
62B10 Statistical aspects of information-theoretic topics
65C60 Computational problems in statistics (MSC2010)
Keywords:
quantiles; moments
Software:
Ox
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adamidis, K.; Loukas, S., A lifetime distribution with decreasing failure rate, Statistics & probability letters, 39, 35-42, (1998) · Zbl 0908.62096
[2] Barreto-Souza, W.; Santos, A.H.S.; Cordeiro, G.M., The beta generalized exponential distribution, Journal of statistical computation and simulation, (2009)
[3] Barreto-Souza, W., Cordeiro, G.M., Simas, A.B., 2008. Some results for beta Fréchet distribution. Preprint: arXiv:0809.1873v1 · Zbl 1216.62018
[4] Cribari-Neto, F.; Zarkos, S.G., Econometric and statistical computing using ox, Computational economics, 21, 277-295, (2003) · Zbl 1047.62119
[5] Doornik, J., Ox: an object-oriented matrix programming language, (2006), Timberlake Consultants and Oxford London
[6] Eugene, N.; Lee, C.; Famoye, F., Beta-normal distribution and its applications, Communications in statistics, theory and methods, 31, 497-512, (2002) · Zbl 1009.62516
[7] Glaser, R.E., Bathtub and related failure rate characterizations, Journal of the American statistical association, 75, 667-672, (1980) · Zbl 0497.62017
[8] Gupta, R.D.; Kundu, D., Generalized exponential distributions, Australian and New Zealand journal of statistics, 41, 173-188, (1999) · Zbl 1007.62503
[9] Hinkley, D., On quick choice of power transformations, The American Statistician, 26, 67-69, (1977)
[10] Jones, M.C., Families of distributions arising from distributions of order statistics, Test, 13, 1-43, (2004) · Zbl 1110.62012
[11] Kus, C., A new lifetime distribution, Computational statistics and data analysis, 51, 4497-4509, (2007) · Zbl 1162.62309
[12] Mudholkar, G.S.; Srivastava, D.K., Exponentiated Weibull family for analysing bathtub failure data, IEEE transactions on reliability, 42, 299-302, (1993) · Zbl 0800.62609
[13] Mudholkar, G.S.; Srivastava, D.K.; Freimer, M., The exponentiated Weibull family, Technometrics, 37, 436-445, (1995) · Zbl 0900.62531
[14] Mudholkar, G.S.; Hutson, A.D., The exponentiated Weibull family: some properties and a flood data application, Communications in statistics, theory and methods, 25, 3059-3083, (1996) · Zbl 0887.62019
[15] Nadarajah, S.; Gupta, A.K., The beta Fréchet distribution, Far east journal of theoretical statistics, 14, 15-24, (2004) · Zbl 1074.62008
[16] Nadarajah, S.; Kotz, S., The beta Gumbel distribution, Mathematical problems in engineering, 10, 323-332, (2004) · Zbl 1068.62012
[17] Nadarajah, S.; Kotz, S., The beta exponential distribution, Reliability engineering and system safety, 91, 689-697, (2005)
[18] Nadarajah, S.; Kotz, S., The exponentiated type distributions, Acta applicandae mathematicae, 92, 97-111, (2006) · Zbl 1128.62015
[19] Nassar, M.M.; Eissa, F.H., On the exponentiated Weibull distribution, Communications in statistics, theory and methods, 32, 1317-1336, (2003) · Zbl 1140.62308
[20] Proschan, F., Theoretical explanation of observed decreasing failure rate, Technometrics, 5, 375-383, (1963)
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.