Geometric-Zero Truncated Poisson Distribution: Properties and Applications

Year 2019, Volume: 32 Issue: 4, 1339 - 1354, 01.12.2019

Yunus Akdogan , Coskun Kus , Hamid Bıdram İsmail Kınacı

https://doi.org/10.35378/gujs.472978

Cited By: 2

Abstract

In this paper, a new discrete distribution is introduced by compounding the
geometric distribution with a zero truncated Poisson distribution, named
geometric-zero truncated Poisson (GZTP) distribution. Some basic properties of
the new distribution, such as the hazard rate function, moments, mode, median,
etc., are studied. We show mathematically and numerically that the hazard rate
function is increasing. The model parameters are estimated by the moment, least
squared error and maximum likelihood methods. A simulation study is performed
to compare the performance of the different estimators in terms of bias and
mean squared error. An application of the new model is also illustrated using
the three real data sets.

Keywords

Compounding, Estimation, Geometric distribution, Zero truncated Poisson distribution

References

• 1. Adamidis, K. and Loukas, S., "A lifetime distribution with decreasing failure rate", Statistics and Probability Letters, 39: 35-42, (1998).
• 2. Akdoğan, Y., Kuş, C., Asgharzadeh, A., Kınacı, I. and Sharafi, F., "Uniform-geometric distribution." Journal of Statistical Computation and Simulation, 86(9): 1754-1770, (2016).
• 3. Bakouch, H.S., Jazi, M.A. and S. Nadarajah, S., "A new discrete distribution." Statistics, 48: 200-240. (2014).
• 4. Barreto-Souza, W., Morais, A.L. and Cordeiro, G.M., "The Weibull-geometric distribution." Journal of Statistical Computation and Simulation, 81(5): 645-657. (2011).
• 5. Braden, B., "Calculating sums of infinite series", The American Mathematical Monthly, 99(7): 649-655, (1992).
• 6. Ferguson, T.S., A course in large sample theory, London: Chapman and Hall. (1996).
• 7. Gomez-Deniz, E., "A new discrete distribution: properties and applications in medical care", Journal of Applied Statistics, 40: 2760-2770, (2013).
• 8. Hemmati, F., Khorram, E. and Rezakhah, S., "A new three-parameter ageing distribution", Journal of Statistical Planning and Inference, 141: 2266-2275. (2011).
• 9. Keilson, J. and H. Gerber, H., "Some results for discrete unimodality." Journal of the American Statistical Association, 66: 386-389, (1971).
• 10. Kemp, A.W., "Classes of discrete lifetime distributions", Communications in Statistics-Theory and Methods 33: 3069-3093, (2004).
• 11. Kuş, C., "A new lifetime distribution", Computational Statistics and Data Analysis, 51: 4497-4509. (2007).
• 12. Lu, W. and Shi, D., "A new compounding life distribution: The Weibull-Poisson distribution." Journal of Applied Statistics, 39: 21-38, (2012).
• 13. Makcutek, J., "A generalization of the geometric distribution and its application in quantitative linguistics", Romanian Rep. Phys, 60: 501-509, (2008).
• 14. Nakagawa, T. and Osaki, S., 1975. "The discrete Weibull distribution”, IEEE Transactions on Reliability, 24: 300-301. (1975).
• 15. Nakagawa, T. and Zhao, X., "Optimization Problems of a Parallel System with a Random Number of Units", IEEE Transactions on Reliability, 61: 543-548, (2012).
• 16. Nekoukhou, V., Alamatsaz, M.H. and Bidram. H., "A discrete analogue of the generalized exponential distribution", Comm. Statist. Theory Methods, 41: 2000-2013, (2012).
• 17. Nekoukhou, V., and H. Bidram, H., "The exponentiated discrete Weibull distribution", SORT, 39: 127-146, (2015).
• 18. Roy, D. and Gupta, R. P., "Classifications of discrete lives", Microelectronics and Reliability, 32(10): 1450-1459, (1992)..
• 19. Shafaei N.M., Rezaei, R.A.H. and Mohtashami, B.G.R., "Some discrete lifetime distributions with bathtub-shaped hazard rate functions", Quality Engineering, 25: 225-236, (2013).
• 20. Steutel, F.W. and Van, K.H., “Infinite Divisibility of Probability Distributions on the Real Line”, New York: Marcel Dekker, (2004).
• 21. Tahmasbi, R. and Rezaei, S., 2008. "A two-parameter lifetime distribution with decreasing failure rate." Computational Statistics and Data Analysis, 52: 3889-3901. (2008).
• 22. Xie, M., and Lai, C.D., "Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function", Reliability Engineering and System Safety, 52: 87-93, (1995).