International Journal of Nonlinear Analysis and Applications
Discrete alpha-power Weibull distribution: Properties and application
Document Type : Research Paper
Authors
Faculty of Science, Mathematics Department, Zagazig University, Zagazig, Egypt
Abstract
A three-parameter discrete analogue of the Alpha-power Weibull distribution (DAPW) is provided in this study. It has established some of its basic distributional and statistical properties. The probability mass function's form, moments, skewness, kurtosis, probability generating function, characteristic function, stress-strength reliability, and order statistics are all examples of this. The unknown parameters are estimated using the maximum likelihood and moments approaches. The bias and mean square error of the maximum likelihood are demonstrated via a simulated exercise. Two datasets are used to demonstrate the model's adaptability.
Keywords
[1] B. Abebe, A discrete Lindely distribution with applications in biological sciences biometrics and biostatistics,
Biomet. Biostatist. Int. J. 7 (2018), no. 1, 48–52.
[2] Z. Ahmed and B. Iqpal, Generalized flexible Weibull extension distribution, Circ. Comput. Sci. 2 (2017), no. 4,
68–75.
[3] G.R. Aryal and C.P. Tsokos, Transmuted Weibull distribution: A generalization of the Weibull probability distribution, Eur. J. Pure Appl. Math. 4 (2011), no. 2, 89–102.
[4] M. Bebbington, C.D. Lai and R. Zitikis, A flexible Weibull extension, Reliab. Engin. Syst. Safety 92 (2007),
719–726.
[5] S. Chakraborty and D. Chakraborty, A new discrete probability distribution with integer support on (−∞, ∞),
Commun. Statist. Theory Method. 45 (2016), no. 2, 492–505.
[6] M. El-Morshedy, M.S. Eliwa and H. Nagy, A new two-parameter exponentiated discrete Lindley distribution:
properties, estimation and applications, J. Appl. Statist. 47 (2020), no. 2, 354–375.
[7] A. Flaih, H. Elsalloukh, E. Mendi and M. Milanova, The exponetiated inverted Weibull distribution, Appl. Math.
Inf. Sci. 6 (2012), no. 2, 167–171.
[8] B. Gnedenko and I. Ushakov, Probabilistic reliability engineering, New York: Wiley, 1995.[9] E. G´omez-D´eniz & E. Calder´ın-Ojeda,The discrete lindley distribution: Properties and applications, J. Statist.
Comput. Simul. 81 (2011), no. 11, 1405–1416.
[10] E. G´omez-D´eniz and E. Calder´ın Ojeda, The compound DGL/Erlang distribution in the collective risk model,
Revist. Methods Cuantt. Para´ la econom ´ la y la empresa, 16 (2013), 121–142.
[11] R.D. Gupta and D. Kundu, A new class of weighted exponential distribution, Statist. 43 (2009), 621–634.
[12] S. Hassan, M. Bakouch, A. Jazi and S. Nadarajah, A new discrete Distribution, Statist. 48 (2014), no. 1, 200–240.
[13] K.C. Kapur and L.R. Lamberson, Reliability in engineering design, John Wiley and Sons, Inc., New York, 1977.
[14] H. Krishna and P. Singh Pundir, Discrete Burr and discrete Pareto distributions, Statist. Meth. 6 (2009), no. 2,
177–188.
[15] S. Kotz and M. Pensky, The stress-strength model and its generalization: Theory and application, World Scientific,
2003.
[16] A. Lai and M.H. Alamatsaz, A discrete inverse Weibull distribution and estimation of its parameters, Statist.
Meth. 7 (2010), no. 2, 121–132.
[17] J.F. Lawless,Statistical models and methods for lifetime data, John Wiley and Sons, Inc., New York, 1982.
[18] A. Mahdavi, D. Kundu, A new method for generating distributions with an application to exponential distribution,
Commun. Statist. Theory Method. 46 (2015), no. 13, 6543–6557.
[19] B.A. Maguire, E. Pearson and A. Wynn, The time intervals between industrial accidents, Biometrica 39 (1952),
no. 1/2, 168–180.
[20] M.O. Mohamed, Inference for reliability and stress-strength for geometric distribution, Sylwan 159 (2015), no. 2,
281–289.
[21] M.O. Mohamed, Estimation of R for geometric distribution under lower record values, J. Appl. Res. Technol. 18
(2020), no. 6, 368–375.
[22] B. Munindra, R.S. Kirshna and J. Junali, A study on two parameters discrete quasi Lindely distribution and its
derived distribution, Int. J. Math. Arch. 6 (2015), no. 12, 149–156.
[23] V. Nekoukhou and H. Bidram, Exponential-discrete generalized exponential distribution: A new compound model,
J. Statist. Theory Appl. 15 (2016), no. 2, 169.
[24] M. Nassar, A. Alzaatreh, M. Mead and O. Abo-Kasem, Alpha power Weibull Distribution: Properties and Applications, Commun. Statist. Theory Method. 46 (2017), no. 20, 10236–10252.
[25] B.A. Para and T.R. Jan, Discretization of Burr-Type III distribution, J. Reliab. Statist. Stud. 7 (2014), no. 2,
87–94.
[26] B.A. Para and T.R. Jan, On discrete three parameter Burr type XII and discrete Lomax distributions and their
applications to model count data from medical science, Biomet. Biostatist. Int. J. 4 (2016), no. 2.
[27] B. A. Para and T. R. Jan, Discrete Inverse Weibull Minimax Distribution: Properties and Applications, J. Statist.
Appl. Prob. 6 (2017), no. 1, 205–218.
[28] M. Pal, M.M. Ali and J. Woo, Exponentiated Weibull distribution, Statistica 66 (2006), no. 2, 139–147.
[29] D. Roy and R.P. Gupta, Classifications of discrete lives, Micro Electron. Reliab. 32 (1992), 1459–1473.
[30] D. Roy, Reliability measures in the discrete bivariate set up and related characterization results for a bivariate
geometric distribution, Multivar. Anal. 46 (1993), 362–373.
[31] D. Roy, The discrete normal distribution, Commun. Statist. Theory Method. 32 (2003), no. 10, 1871–1883.
[32] D. Roy, Discrete Rayleigh distribution, IEEE Trans. Reliab. 53 (2004), no. 2, 255–260.
[33] M. Shafaei Nooghabi, G.R. Mohtashami Borzadaran and A. Hamid Rezaei Roknabadi, Discrete modified Weibull,
METRON, 69 (2011), 207.
[34] S.K. Sinha, Reliability and life testing, Wiley Eastern Limited, New Delhi, 1986.[35] W. Barreto-Souza, A.L. de Morais and M. Gauss Cordeiro, The Weibull-geometric distribution, J. Statist. Comput.
Simul. 81 (2011), no. 5, 645–657.
Biomet. Biostatist. Int. J. 7 (2018), no. 1, 48–52.
[2] Z. Ahmed and B. Iqpal, Generalized flexible Weibull extension distribution, Circ. Comput. Sci. 2 (2017), no. 4,
68–75.
[3] G.R. Aryal and C.P. Tsokos, Transmuted Weibull distribution: A generalization of the Weibull probability distribution, Eur. J. Pure Appl. Math. 4 (2011), no. 2, 89–102.
[4] M. Bebbington, C.D. Lai and R. Zitikis, A flexible Weibull extension, Reliab. Engin. Syst. Safety 92 (2007),
719–726.
[5] S. Chakraborty and D. Chakraborty, A new discrete probability distribution with integer support on (−∞, ∞),
Commun. Statist. Theory Method. 45 (2016), no. 2, 492–505.
[6] M. El-Morshedy, M.S. Eliwa and H. Nagy, A new two-parameter exponentiated discrete Lindley distribution:
properties, estimation and applications, J. Appl. Statist. 47 (2020), no. 2, 354–375.
[7] A. Flaih, H. Elsalloukh, E. Mendi and M. Milanova, The exponetiated inverted Weibull distribution, Appl. Math.
Inf. Sci. 6 (2012), no. 2, 167–171.
[8] B. Gnedenko and I. Ushakov, Probabilistic reliability engineering, New York: Wiley, 1995.[9] E. G´omez-D´eniz & E. Calder´ın-Ojeda,The discrete lindley distribution: Properties and applications, J. Statist.
Comput. Simul. 81 (2011), no. 11, 1405–1416.
[10] E. G´omez-D´eniz and E. Calder´ın Ojeda, The compound DGL/Erlang distribution in the collective risk model,
Revist. Methods Cuantt. Para´ la econom ´ la y la empresa, 16 (2013), 121–142.
[11] R.D. Gupta and D. Kundu, A new class of weighted exponential distribution, Statist. 43 (2009), 621–634.
[12] S. Hassan, M. Bakouch, A. Jazi and S. Nadarajah, A new discrete Distribution, Statist. 48 (2014), no. 1, 200–240.
[13] K.C. Kapur and L.R. Lamberson, Reliability in engineering design, John Wiley and Sons, Inc., New York, 1977.
[14] H. Krishna and P. Singh Pundir, Discrete Burr and discrete Pareto distributions, Statist. Meth. 6 (2009), no. 2,
177–188.
[15] S. Kotz and M. Pensky, The stress-strength model and its generalization: Theory and application, World Scientific,
2003.
[16] A. Lai and M.H. Alamatsaz, A discrete inverse Weibull distribution and estimation of its parameters, Statist.
Meth. 7 (2010), no. 2, 121–132.
[17] J.F. Lawless,Statistical models and methods for lifetime data, John Wiley and Sons, Inc., New York, 1982.
[18] A. Mahdavi, D. Kundu, A new method for generating distributions with an application to exponential distribution,
Commun. Statist. Theory Method. 46 (2015), no. 13, 6543–6557.
[19] B.A. Maguire, E. Pearson and A. Wynn, The time intervals between industrial accidents, Biometrica 39 (1952),
no. 1/2, 168–180.
[20] M.O. Mohamed, Inference for reliability and stress-strength for geometric distribution, Sylwan 159 (2015), no. 2,
281–289.
[21] M.O. Mohamed, Estimation of R for geometric distribution under lower record values, J. Appl. Res. Technol. 18
(2020), no. 6, 368–375.
[22] B. Munindra, R.S. Kirshna and J. Junali, A study on two parameters discrete quasi Lindely distribution and its
derived distribution, Int. J. Math. Arch. 6 (2015), no. 12, 149–156.
[23] V. Nekoukhou and H. Bidram, Exponential-discrete generalized exponential distribution: A new compound model,
J. Statist. Theory Appl. 15 (2016), no. 2, 169.
[24] M. Nassar, A. Alzaatreh, M. Mead and O. Abo-Kasem, Alpha power Weibull Distribution: Properties and Applications, Commun. Statist. Theory Method. 46 (2017), no. 20, 10236–10252.
[25] B.A. Para and T.R. Jan, Discretization of Burr-Type III distribution, J. Reliab. Statist. Stud. 7 (2014), no. 2,
87–94.
[26] B.A. Para and T.R. Jan, On discrete three parameter Burr type XII and discrete Lomax distributions and their
applications to model count data from medical science, Biomet. Biostatist. Int. J. 4 (2016), no. 2.
[27] B. A. Para and T. R. Jan, Discrete Inverse Weibull Minimax Distribution: Properties and Applications, J. Statist.
Appl. Prob. 6 (2017), no. 1, 205–218.
[28] M. Pal, M.M. Ali and J. Woo, Exponentiated Weibull distribution, Statistica 66 (2006), no. 2, 139–147.
[29] D. Roy and R.P. Gupta, Classifications of discrete lives, Micro Electron. Reliab. 32 (1992), 1459–1473.
[30] D. Roy, Reliability measures in the discrete bivariate set up and related characterization results for a bivariate
geometric distribution, Multivar. Anal. 46 (1993), 362–373.
[31] D. Roy, The discrete normal distribution, Commun. Statist. Theory Method. 32 (2003), no. 10, 1871–1883.
[32] D. Roy, Discrete Rayleigh distribution, IEEE Trans. Reliab. 53 (2004), no. 2, 255–260.
[33] M. Shafaei Nooghabi, G.R. Mohtashami Borzadaran and A. Hamid Rezaei Roknabadi, Discrete modified Weibull,
METRON, 69 (2011), 207.
[34] S.K. Sinha, Reliability and life testing, Wiley Eastern Limited, New Delhi, 1986.[35] W. Barreto-Souza, A.L. de Morais and M. Gauss Cordeiro, The Weibull-geometric distribution, J. Statist. Comput.
Simul. 81 (2011), no. 5, 645–657.
)
Volume 13, Issue 2
July 2022Pages 1305-1317
- Receive Date: 02 February 2022
- Revise Date: 25 February 2022
- Accept Date: 10 March 2022