A Study of Some Properties and Goodness-of-Fit of a Gompertz-Rayleigh Model

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Fadimatu Bawuro Mohammed
Kabiru Ahmed Manju
Umar Kabir Abdullahi
Makama Musa Sani
Samson Kuje


The Rayleigh was obtained from the amplitude of sound resulting from many important sources by Rayleigh. It is continuous probability distribution with a wide range of applications such as in life testing experiments, reliability analysis, applied statistics and clinical studies. However, it is not flexible enough for modeling heavily skewed datasets as compared to compound distributions. In this paper, we introduce a new extension of the Rayleigh distribution by using a Gompertz-G family of distributions. This paper defines and studies a three-parameter distribution called “Gompertz-Rayleigh distribution”. Some properties of the proposed distribution are derived and discussed comprehensively in this paper and the three parameters are estimated using the method of maximum likelihood estimation. The goodness-of-fit of the proposed distribution is also evaluated by fitting it in comparison with some other existing distributions using a real life data.

Rayleigh distribution, Gompertz-Rayleigh distribution, properties, parameters, method of maximum likelihood estimation and goodness-of-fit.

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How to Cite
Mohammed, F. B., Manju, K. A., Abdullahi, U. K., Sani, M. M., & Kuje, S. (2020). A Study of Some Properties and Goodness-of-Fit of a Gompertz-Rayleigh Model. Asian Journal of Probability and Statistics, 9(2), 18-31. https://doi.org/10.9734/ajpas/2020/v9i230223
Original Research Article


Rayleigh J. On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. Phil. & Manag. 1980;10:73–78.

Siddiqui MM. Some problems connected with Rayleigh distributions. Journal of Research Nat. Bureau of the Stand. 1962;60D:167–174.

Hirano K. Rayleigh distributions. New York: John Wiley; 1986.

Howlader HA, Hossain A. On Bayesian estimation and prediction from Rayleigh distribution based on type-II censored data. Comm. Stat.: Theo & Meth. 1995;24(9):2249-2259.

Kundu D, Raqab MZ. Generalized Rayleigh distribution: Different methods of estimations. Comp. Stat. & Data Anal. 2005;49:187–200.

Abdel-Hady DH. Bivariate generalized Rayleigh distribution. J. of Appl. Sci. Res. 2013;9(9):5403-5411.

Merovci F. The transmuted Rayleigh distribution. Australian J. of Stat. 2013;22(1):21–30.

Yahaya A, Alaku AY. On generalized Weibull-Rayleigh distribution: Its properties and applications. ATBU J. of Sci., Techn. and Edu. 2018;5(1):217-231.

Merovci F, Elbatal I. Weibull Rayleigh distribution: Theory and applications. Appl. Math. & Inf. Sci. 2015;9(5):1-11.

Yahaya A, Ieren TG. A note on the transmuted Weibull-Rayleigh distribution. Edited Proceedings of 1st Int. Conf. of Nigeria Stat. Soc. 2017;1:7-11.

Ahmad A, Ahmad SP, Ahmed A. Transmuted inverse Rayleigh distribution: A generalization of the inverse Rayleigh distribution. Math. Theo. & Mod. 2015;4(7):90-98.

Oguntunde PE, Balogun OS, Okagbue HI, Bishop SA. The Weibull-exponential distribution: Its properties and applications. J. of Appl. Sci. 2015;15(11):1305-1311.

Afify MZ, Yousof HM, Cordeiro GM, Ortega EMM, Nofal ZM. The Weibull Frechet distribution and its applications. J. of Appl. Stat. 2016;1-22.

Ieren TG, Kuhe AD. On the properties and applications of Lomax-exponential distribution. Asian J. of Prob. & Stat. 2018;1(4):1-13.

Ieren TG, Oyamakin SO, Chukwu AU. Modeling lifetime data with Weibull-Lindley distribution. Biomet. & Biostat. Int. J. 2018;7(6):532‒544.

Koleoso PO, Chukwu AU, Bamiduro TA. A three-parameter Gompertz-Lindley distribution: Its properties and applications. J. of Math. Theo. & Mod. 2019;9(4):29-42.

Ieren TG, Koleoso PO, Chama AF, Eraikhuemen IB, Yakubu N. A Lomax-inverse Lindley distribution: Model, properties and applications to lifetime data. J. of Adv. in Math. & Comp. Sci. 2019;34(3-4):1-28.

Umar AA, Eraikhuemen IB, Koleoso PO, Joel J, Ieren TG. On the properties and applications of a transmuted Lindley-exponential distribution. Asian J. of Prob. & Stat. 2019;5(3):1-13.

Ieren TG, Kromtit FM, Agbor BU, Eraikhuemen IB, Koleoso PO. A power Gompertz distribution: Model, properties and application to bladder cancer data. Asian Res. J. of Math. 2019;15(2):1-14.

Alizadeh M, Cordeiro GM, Bastos Pinho LG, Ghosh I. The Gompertz-G family of distributions. J. of Stat. The and Pract. 2017;11(1):179–207.

Hyndman RJ, Fan Y. Sample quantiles in statistical packages. The American Statistician. 1996;50(4): 361-365.

Kenney JF, Keeping ES. Mathematics of statistics. 3 Edn, Chapman & Hall Ltd, New Jersey; 1962.

Moors JJ. A quantile alternative for kurtosis. J. of the Royal Stat. Soc.: Series D. 1988;37:25–32.

Chen G, Balakrishnan N. A general purpose approximate goodness-of-fit test. Journal of Quality Technology. 1995;27:154–161.

Afify AZ, Aryal G. The Kummaraswamy exponentiated Frechet distribution. J. of Data Sci. 2016;6:1-19.

Barreto-Souza WM, Cordeiro GM, Simas AB. Some results for beta Frechet distribution. Comm. in Stat.: Theo. & Meth. 2011;40:798-811.

Bourguignon M, Silva RB, Cordeiro GM. The Weibull-G family of probability distributions. J. of Data Sci. 2014;12:53-68.

Ieren TG, Yahaya A. The Weimal distribution: Its properties and applications. J. of the Nigeria Ass. of Math. Physics. 2017;39:135-148.

Yahaya A, Ieren TG. On transmuted Weibull-exponential distribution: Its properties and applications. Nigerian J. of Sci. Res. 2017;16(3):289-297.

Smith RL, Naylor JC. A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. J. of Appl. Stat. 1987;36:358-369.