Abstract

In this study, we have launched a new two-parameter probability model called the New Lindley-Rayleigh distribution. The proposed model accommodates unimodal and bathtub, and a broad variety of monotone failure rates. Some statistical and mathematical properties of this distribution are discussed. Four widely used estimation methods are employed to estimate the model parameters namely maximum likelihood estimators (MLE), leastsquare (LSE) and Cramer-Von-Mises (CVM) methods. By using the maximum likelihood estimate we have constructed the asymptotic confidence interval for the model parameters. The potentiality of the proposed distribution is revealed by using a real dataset, where the proposed distribution provided better fit in comparison with some other lifetime distributions. The importance of the proposed distribution is illustrated by using a real dataset, and found that it provides a better fitting in comparison with other lifetime distributions.

Keywords

References

[1] Rayleigh, L., “On the stability or instability of certain fluid motions,” Proceedings of London Mathematical Society, vol. 11, pp. 57-70, 1880.
[2] Dyer, D.D. and Whisenand, C.W., “Best linear unbiased estimator of the parameter of the Rayleigh distribution”, IEEE Transaction on Reliability, vol. 22, pp. 27-34, 1973.
[3] Voda, V. G., “Note on the truncated Rayleigh variate”, Revista Colombiana de Matematicas, vol. 9, pp. 1–7, 1975.
[4] Bhattacharya, S.K. and Tyagi, R. K., “Bayesian survival analysis based on the Rayleigh model,” Trabajos de Estadistica, vol. 5, pp. 81- 92, 1990.
[5] Fernandez, A.J., “Bayesian estimation and prediction based on Rayleigh sample quantiles”. Quality & Quantity, vol. 44, p. 1239-1248, 2010.
[6] Gomes, A.E., da-Silva, A.Q., Cordeiro, G.M. and Ortega, E.M.M., “A new lifetime model: the Kumaraswamy generalized Rayleigh distribution”, Journal of Statistical Computation and Simulation, vol. 84, pp. 280-309, 2014.
[7] MirMostafaee, S. M. T. K., Mahdizadeh, M., & Lemonte, A. J., “The Marshall–Olkin extended generalized Rayleigh distribution: Properties and applications”. Communications in Statistics-Theory and Methods, vol. 46(2), pp. 653-671, 2017.
[8] Iriate, Y.A., Vilca, F., Varela, H. and Gomez, H.W., “Slashed generalized Rayleigh distribution”, Communications in Statistics - Theory and Methods, vol. 46, p. 4686-4699, 2017.
[9] Cakmakyapan, S., & Ozel, G., “New Lindley-Rayleigh distribution with application to lifetime data”. Journal of Reliability and Statistical Studies, vol. 11(2), 2018.
[10] Iriarte, Y. A., Castillo, N. O., Bolfarine, H., & Gómez, H. W., “Modified slashed-Rayleigh distribution”. Communications in Statistics- Theory and Methods, vol. 47(13), pp. 3220-3233, 2018.
[11] Biçer, H. D., “Properties and Inference for a New Class of Generalized Rayleigh Distributions with an Application”. Open Mathematics, vol. 17(1), pp. 700-715, 2019.
[12] Lindley, D.V., “Fiducial distributions and Bayes’ theorem,” Journal of the Royal Statistical Society Series B, vol. 20, pp. 102-107, 1958.
[13] Ghitany, M.E., Atieh, B., & Nadarajah, S., “Lindley distribution and its application”. Math. Comput. Simul. Vol. 78, pp. 493–506, 2008.
[14] Ristić, M. M., & Balakrishnan, N., “The gamma-exponentiated exponential distribution”. Journal of Statistical Computation and Simulation, vol. 82(8), pp. 1191-1206, 2012.
[15] Moors, J. J. A., “A quantile alternative for kurtosis”. Journal of the Royal Statistical Society: Series D (The Statistician), vol. 37(1), pp. 25-32, 1988.
[16] Swain, J. J., Venkatraman, S. & Wilson, J. R., ‘Least-squares estimation of distribution functions in johnson’s translation system’, Journal of Statistical Computation and Simulation vol. 29(4), pp.271–297, 1988.
[17] Macdonald, P. D. M., “Comments and Queries Comment on “An Estimation Procedure for Mixtures of Distributions” by Choi and Bulgren.” Journal of the Royal Statistical Society: Series B (Methodological), vol. 33(2), pp. 326-329, 1971.
[18] Hinkley, D., “On quick choice of power transformations.” Journal of the Royal Statistical Society, Series (c), Applied Statistics, vol. 26, pp. 67-69, 1977.
[19] Venables, W. N., Smith, D. M. and R Development Core Team. An Introduction to R, R Foundation for Statistical Computing, Vienna,Austria, 2020. ISBN 3-900051-12-7. URL http://www.r-project.org.
[20] Kumar, V. and Ligges, U., reliaR : A package for some probability distributions, 2011. http://cran.rproject. org/web/packages/reliaR/index.html.
[21] Kundu, D., and Raqab, M.Z., “Generalized Rayleigh Distribution: Different Methods of Estimation,” Computational Statistics and Data Analysis, vol. 49, pp. 187-200, 2005.
[22] Smith, R.M. and Bain, L.J., “An exponential power life-test distribution,” Communications in Statistics, vol. 4, pp. 469-481, 1975.
[23] Murthy, D.N.P., Xie, M. and Jiang, R., Weibull Models, Wiley, New York, 2003.
[24] Nadarajah, S. and Haghighi, F., “An extension of the exponential distribution.” Statistics, vol. 45(6), pp. 543-558, 2011.