ON THE PROPERTIES AND APPLICATIONS OF A NEW EXTENSION OF EXPONENTIATED RAYLEIGH DISTRIBUTION
Statistical distributions already in existence are not the most appropriate model that adequately describes real-life data such as those obtained from experimental investigations. Therefore, there are needs to come up with their extended forms to give substitutive adaptable models. By adopting the method of Transformed-Transformer family of distributions, an extension of Exponentiated Rayleigh distribution titled Gompertz- Exponentiated Rayleigh (GOM-ER) distribution was proposed and proved to be valid. Some properties of the new distribution including random number generator, quartiles, distribution of smallest and largest order statistics, reliability function, hazard rate function, cumulative or integrated hazard function, odds function, non-central moments, moment generating function, mean, variance and entropy measures were derived. Using the methods of maximum likelihood and maximum product of spacing, the four unknown parameters were estimated. Shapes of the hazard function depicts that GOM-ER is a distribution that is strictly increasing while those of the PDF depicts that GOM-ER can be skewed or symmetrical. Two datasets were fitted to determine the flexibility of GOM-ER. Simulation study evaluates the consistency, accuracy and unbiasedness of the GOM-ER parameter estimates obtained from the two frequentist estimation methods adopted.
Abd-Elfattah, A. (2011). Goodness of fit test for the generalized rayleigh distribution with unknown parameters. Journal of Statistical Computation and Simulation, 81(3):357–366. DOI: https://doi.org/10.1080/00949650903348155
Al-Mofleh, H. and Afify, A. Z. (2019). A generalization of ramos-louzada distribution: Properties and estimation.arXiv preprint arXiv: 1912.08799.
Alizadeh, M., Cordeiro, G. M., Pinho, L. G. B., and Ghosh, I. (2017). The gompertz-g family of distributions. Journal of Statistical Theory and Practice, 11(1):179–207. DOI: https://doi.org/10.1080/15598608.2016.1267668
Alzaatreh, A., Lee, C., and Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71(1):63–79. DOI: https://doi.org/10.1007/s40300-013-0007-y
Alzaghal, A., Famoye, F., and Lee, C. (2013). Exponentiated tx family of distributions with some applications. International Journal of Statistics and Probability, 2(3):31. DOI: https://doi.org/10.5539/ijsp.v2n3p31
Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, pages 171–178.
Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistica, 46(2):199–208.
Bhat, A. and Ahmad, S. (2020). Pak. j. statist. 2020 vol. 36 (3), 225-250 a new generalization of rayleigh distri- bution: Properties and applications. Pak. J. Statist, 36(3):225–250.
Bourguignon, M., Silva, R. B., and Cordeiro, G. M. (2014). The weibull-g family of probability distributions. Journal of Data Science, 12(1):53–68. DOI: https://doi.org/10.6339/JDS.201401_12(1).0004
Cheng, R. and Amin, N. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society: Series B (Methodological), 45(3):394–403. DOI: https://doi.org/10.1111/j.2517-6161.1983.tb01268.x
Cordeiro, G. M. and de Castro, M. (2011). A new family of generalized distributions. Journal of statistical computation and simulation, 81(7):883–898. DOI: https://doi.org/10.1080/00949650903530745
Eghwerido, J., Zelibe, S., and Efe-Eyefia, E. (2020). Gompertz-alpha power iverted exponential distribution: Properties and applications. Thailand Statistician, 18(3):319–332.
Eugene, N., Lee, C., and Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and methods, 31(4):497–512. DOI: https://doi.org/10.1081/STA-120003130
Falgore, J. Y. and Doguwa, S. I. (2020). The inverse lomax-g family with application to breaking strength data. Asian Journal of Probability and Statistics, pages 49–60. DOI: https://doi.org/10.9734/ajpas/2020/v8i230204
Ghazal, M. and Hasaballah, H. (2017). Exponentiated rayleigh distribution: A bayes study using mcmc approach based on unified hybrid censored data. Journal of Advances in Mathematics, 12(12):6863–6880. DOI: https://doi.org/10.24297/jam.v12i12.4599
Gupta, R. D. and Kundu, D. (1999). Theory & methods: Generalized exponential distributions. Australian & New Zealand Journal of Statistics, 41(2):173–188. DOI: https://doi.org/10.1111/1467-842X.00072
Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika,36(1/2):149–176. DOI: https://doi.org/10.1093/biomet/36.1-2.149
Jones, M. (2009). Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages.Statistical Methodology, 6(1):70–81. DOI: https://doi.org/10.1016/j.stamet.2008.04.001
Kundu, D. and Raqab, M. Z. (2005). Generalized rayleigh distribution: different methods of estimations. Computational statistics & data analysis, 49(1):187–200. DOI: https://doi.org/10.1016/j.csda.2004.05.008
Madi, M. and Raqab, M. (2009). Bayesian analysis for the exponentiated rayleigh distribution. Metron Int. J.Statistics, 67:269–288.
Mahmoud, M. and Ghazal, M. (2017). Estimations from the exponentiated rayleigh distribution based on gener- alized type-ii hybrid censored data. Journal of the Egyptian Mathematical Society, 25(1):71–78. DOI: https://doi.org/10.1016/j.joems.2016.06.008
Marshall, A. W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and weibull families. Biometrika, 84(3):641–652. DOI: https://doi.org/10.1093/biomet/84.3.641
Mohammed, F. B., Manju, K. A., Abdullahi, U. K., Sani, M. M., and Kuje, S. (2020). A study of some properties and goodness-of-fit of a gompertz-rayleigh model. Asian Journal of Probability and Statistics, pages 18–31. DOI: https://doi.org/10.9734/ajpas/2020/v9i230223
Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated weibull family for analyzing bathtub failure-rate data. IEEE transactions on reliability, 42(2):299–302. DOI: https://doi.org/10.1109/24.229504
Nofal, Z. M., Afify, A. Z., Yousof, H. M., and Cordeiro, G. M. (2017). The generalized transmuted-g family of distributions. Communications in Statistics-Theory and Methods, 46(8):4119–413 DOI: https://doi.org/10.1080/03610926.2015.1078478
Oguntunde, P. E., Adejumo, A. O., Khaleel, M. A., Okagbue, H. I., and Odetunmibi, O. A. (2018). The logistic inverse exponential distribution: Basic structural properties and application. World Congress on Engineering. DOI: https://doi.org/10.1007/978-981-13-0746-1_8
Pathak, A. and Chaturvedi, A. (2014). Estimation of the reliability function for two-parameter exponentiated rayleigh or burr type x distribution. Statistics, Optimization & Information Computing, 2(4):305–322. DOI: https://doi.org/10.19139/soic.v2i4.36
Ranneby, B. (1984). The maximum spacing method. An estimation method related to the maximum likelihood method. Scandinavian Journal of Statistics, pages 93–112.
Rashwan, N. I. (2016). A note on kumaraswamy exponentiated rayleigh distribution. Journal of Statistical Theory and Applications, 15(3):286–295. DOI: https://doi.org/10.2991/jsta.2016.15.3.8
Rényi, A. (1961). On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics. The Regents of the University of California.
Sen, S., Afify, A. Z., Al-Mofleh, H., and Ahsanullah, M. (2019). The quasi xgamma-geometric distribution with application in medicine. Filomat, 33(16):5291–5330. DOI: https://doi.org/10.2298/FIL1916291S
Voda˘, V. G. (1976). Inferential procedures on a generalized rayleigh variate. i. Aplikace matematiky, 21(6):395–412. DOI: https://doi.org/10.21136/AM.1976.103663
Weibull, W. (1939). A statistical theory of the strength of material. IEEE transactions on reliability, (151).
Yahaya, A. and Ieren, T. G. (2017). A note on the transmuted weibull-rayleigh distribution. In Edited Proceedings of 1st Int. Conf. of Nigeria Stat. Soc, volume 1, pages 7–11
Copyright (c) 2021 FUDMA JOURNAL OF SCIENCES
This work is licensed under a Creative Commons Attribution 4.0 International License.
FUDMA Journal of Sciences