Gumbel-Geometric Distribution: Properties and Applications

Abstract

A three-parameter generalization of the Gumbel distribution, which we call Gumbel-geometric distribution, is defined and investigated. The shape of the density and hazard function is examined and discussed. Explicit expressions for the moment generating function, the characteristics function and the rth order statistic are obtained. Other properties of the distribution are also discussed. The method of maximum likelihood is proposed for the estimation of the parameter of the model and discussed. A simulation experiment is carried out to examine the asymptotic properties of the distribution. The result shows that the MSE decreases to zero as while the bias either increases or decreases (depending on the sign) for each of the parameters. The new distribution is applied to two datasets and compared to some existing generalization to illustrate its flexibility.  

Keywords

• Gumbel distribution
• Extreme values
• Reliability
• Distribution generator
• Geometric distribution

References

1. Gumbel, E.J., Statistics of Extremes. Columbia University Press(1958).
2. Kotz, S. and S. Nadarajah, Extreme Value Distributions: Theory and Applications. London: Imperial College Press (2000).
3. Gómez, Y.M., H. Bolfarine, and H.W. Gómez, "Gumbel distribution with heavy tails and applications to environmental data". Mathematics and Computers in Simulation: (2018).
4. Nadarajah, S., "The exponentiated Gumbel distribution with climate application". Environmetrics, 17: 13-23 (2006).
5. Eugene, N., C. Lee, and F. Famoye, "Beta-Normal distribution and Its applications". Communications in Statistics - Theory and Methods, 31(4): 497-512 (2002).
6. Nadarajah, S. and S. Kotz, "The beta Gumbel distribution". Mathematical Problems in Engineering, 4: 323-332 (2004).
7. Cordeiro, G.M., S. Nadarajah, and E.M. Ortega, "The Kumaraswamy Gumbel distribution.". Statistical Methods and Applications, 21(2): 139-168 (2012).
8. Kumaraswamy, P., "A Generalized Probability Density Function for Doubly Bounded Random Process". Journal of Hydrology, 46: 79-88 (1980).

9. Jones, M.C., "Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages.". Statistical Methodology, 6: 70–81 (2009).
10. Gupta, R.D. and D. Kundu, "Exponentiated Exponential Family: An Alternative to Gamma and Weibull Distributions". Biometrical Journal, 43(1): 117-130 (2001).
11. Okasha, H.M., "A New Family of Topp and Leone Geometric Distribution with Reliability Applications". Journal of Failure Analysis and Prevention: 1-13 (2017).
12. Gilchrist, W.G., Statistical modelling with quantile functions Boca Raton, LA: Chapman &Hall/CRC(2001).
13. Prudnikov, A.P., Y.A. Brychkov, and O.I. Marichev, Integrals and Series: Elementary functions. Vol. 1. Amsterdam: Gordon and Breach(1986).
14. Shahbaz, M.Q., et al., Ordered Random Variables: Theory and Applications. Paris: Atlantis press(2016).
15. Thukral, A.K., "Factorials of real negative and imaginary numbers - A new perspective". SpringerPlus, 2(658): (2014).
16. Akinsete, A., F. Famoye, and C. Lee, "The beta-Pareto distribution". Statistics, 42(6): 547-563 (2008).
17. Nadarajah, S., "The beta exponential distribution". Reliability Eng. Syst. Safety, 91: 689-697 (2006).
18. Hinkley, D., "On quick choice of power transformations". The American Statistician, 26: 67-69 (1977).
19. Changery, M.J., "Historical Extreme Winds for the United States: Atlantic and Gulf of Mexico Coastlines", North Carolina, U.S. Nuclear Regulatory Commission, (1982).
20. Andrade, T., et al., "The Exponentiated Generalized Gumbel Distribution". Revista Colombiana de Estadística, 38(1): 123-143 (2015).
21. Kinnison, R.R., Applied Extreme Value Statistics. 1983, Pacific Northwest Laboratory: Washington.