A new reciprocal Rayleigh extension: properties, copulas, different methods of estimation and a modified right-censored test for validation

Haitham M. Yousof; M. Masoom Ali; Hafida Goual; Mohamed Ibrahim

doi:10.21307/stattrans-2021-029

A new reciprocal Rayleigh extension: properties, copulas, different methods of estimation and a modified right-censored test for validation

Haitham M. Yousof Department of Statistics, Mathematics and Insurance, Faculty of Commerce, Benha University, Benha 13518, Egypt ORCID:https://orcid.org/0000-0003-4589-4944 , M. Masoom Ali Department of Mathematical Sciences Ball State University, Muncie, Indiana 47306, USA ORCID:https://orcid.org//0000-0002-0120-9442 , Hafida Goual Laboratory of Probability and Statistics, University of Badji Mokhtar, Annaba, Algeria ORCID:https://orcid.org/0000-0003-4932-4332 , Mohamed Ibrahim Department of Applied, Mathematical and Actuarial Statistics, Faculty of Commerce, Damietta University, Damietta, Egypt ORCID:https://orcid.org/0000-0003-4893-9669 Statistics in Transition new series, vol. 22, 2021, 3, pages: 99–121 Published online: 6 September 2021 DOI 10.21307/stattrans-2021-029

1099 Views 27 Downloads

ARTICLE

(English) PDF

ABSTRACT

In this article, a new reciprocal Rayleigh extension called the Xgamma reciprocal Rayleigh model is defined and studied. The relevant statistical properties are derived, and the useful results related to the convexity and concavity are addressed. We discussed the estimation of the parameters using different estimation methods such as the maximum likelihood estimation method, the ordinary least squares estimation method, the weighted least squares estimation method, the Cramer-Von-Mises estimation method, and the bootstrapping method. A simulation study was conducted to assess the performances of the proposed estimation methods are investigated through a simulation study. Many bivariate and multivariate type model have also been derived based on Farlie-Gumbel-Morgenstern copula, the Clayton copula, Renyi’s entropy copula and the Ali-Mikhail-Haq copula. A modified Nikulin-Rao-Robson test for right-censored validation is applied to a censored real data set.

KEYWORDS

Xgamma model, reciprocal Rayleigh model, simulations, bootstrapping, Farlie Gumbel Morgenstern copula, least squares, Cramer-Von-Mises, bootstrapping, Ali-Mikhail- Haq copula, convexity, concavity

REFERENCES

Alizadeh, M., Ghosh, I., Yousof, H. M., Rasekhi, M. and Hamedani G. G., (2017). The generalized odd generalized exponential family of distributions: properties, characterizations and applications, J. Data Sci., Vol. 15, pp. 443–466.

Alizadeh, M., Jamal, F., Yousof, H. M., Khanahmadi, M. and Hamedani, G. G., (2020). Flexible Weibull generated family of distributions: characterizations, mathematical properties and applications. University Politehnica of Bucharest Scientific Bulletin- Series A-Applied Mathematics and Physics, Vol. 82, pp. 145–150.

Alizadeh, M., Rasekhi, M., Yousof, H. M. and Hamedani G. G., (2018). The transmuted Weibull G family of distributions. Hacettepe Journal of Mathematics and Statistics, Vol. 47, pp. 1–20.

Alizadeh, M., Yousof, H. M. Rasekhi, M. and Altun, E., (2019). The odd log-logistic Poisson-G Family of distributions, Journal of Mathematical Extensions, Vol. 12, pp. 81–104.

Aryal, G. R., Yousof, H. M., (2017). The exponentiated generalized-G Poisson family of distributions. Economic Quality Control, 32, pp. 1–17.

Bagdonavicius, V., Nikulin, M., (2011a). Chi-squared Goodness-of-fit Test for Right Censored Data. International Journal of Applied Mathematics and Statistics, Vol. 24, pp. 30–50.

Bagdonavicius, V., Nikulin, M., (2011b). Chi-squared tests for general composite hypotheses from censored samples. Comptes Rendus de l'académie des Sciences de Paris, Mathématiques, Vol. 349, pp. 219–223.

Brito, E., Cordeiro, G. M., Yousof, H. M., Alizadeh, M. and Silva, G. O., (2017). Topp- Leone Odd Log-Logistic Family of Distributions, Journal of Statistical Computation and Simulation, Vol. 87, pp. 3040–3058.

Cordeiro, G. M., Altun, E., Korkmaz, M. C., Pescim, R. R., Afify, A. Z. and Yousof, H. M., (2020). The xgamma Family: Censored Regression Modelling and Applications. Revista Colombiana de Estadística, Vol. 18, pp. 593–612.

Cordeiro, G. M., Yousof, H. M., Ramires, T. G. and Ortega, E. M. M., (2018). The Burr XII system of densities: properties, regression model and applications. Journal of Statistical Computation and Simulation, Vol. 88, pp. 432–456.

Chakraborty, S., Handique, L., Altun, E. and Yousof, H. M., (2018). A new statistical model for extreme values: mathematical properties and applications. International Journal of Open Problems in Computer Science and Mathematics, Vol. 12, pp. 1–18.

Elgohari, H., Yousof, H. M., (2020a). A Generalization of Lomax Distribution with Properties, Copula and Real Data Applications. Pakistan Journal of Statistics and Operation Research, Vol. 16, pp. 697–711.

Elgohari, H., Yousof, H. M., (2020b). New Extension of Weibull Distribution: Copula, Mathematical Properties and Data Modeling. Statistics, Optimization & Information Computing, Vol. 8, pp. 972–993.

El-Morshedy, M., Alshammari, F. S., Hamed, Y. S., Eliwa, M. S., Yousof, H. M., (2021). A new family of continuous probability distributions. Entropy, Vol. 23, 194. https://doi.org/10.3390/e23020194

Fréchet, M., (1927). Sur la loi de probabilité de lécart maximum. Ann. Soc. Pol. Math., Vol. 6, pp. 93–116.

Gradshteyn, I. S., Ryzhik, I. M., (2002). Table of integrals, series, and products. San Diego, CA: Academic Press.

Goual, H., Yousof, H. M. and Ali, M. M., (2019). Validation of the odd Lindley exponentiated exponential by a modified goodness of fit test with applications to censored and complete data. Pakistan Journal of Statistics and Operation Research, Vol. 15, pp. 745–771.

Goual, H., Yousof, H. M., (2019). Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test. Journal of Applied Statistics, Vol. 47, pp. 1–32.

Goual, H., Yousof, H. M., and Ali, M. M., (2020). Lomax inverse Weibull model: properties, applications, and a modified Chi-squared goodness-of-fit test for validation. Journal of Nonlinear Sciences & Applications (JNSA), Vol. 13, pp. 330– 353.

Gusmao, F. R. S., Ortega, E. M. M. and Cordeiro, G. M., (2011). The generalized inverse Weibull distribution. Statistical Papers, Vol. 52, pp. 591–619.

Haq, M. A., Yousof, H. M. and Hashmi, S., (2017). A New Five-Parameter Fréchet Model for Extreme Values. Pakistan Journal of Statistics and Operation Research, Vol. 13(3), pp. 617–632.

Hamedani G. G., Altun, E, Korkmaz, M. C., Yousof, H. M. and Butt, N. S., (2018). A new extended G family of continuous distributions with mathematical properties, characterizations and regression modeling. Pakistan Journal of Statistics and Operation Research, Vol. 14, pp. 737–758.

Hamedani G. G., Rasekhi, M., Najibi, S. M., Yousof, H. M. and Alizadeh, M., (2019). Type II general exponential class of distributions. Pakistan Journal of Statistics and Operation Research , Vol. 15, pp. 503–523.

Hamedani G. G., Yousof, H. M., Rasekhi, M., Alizadeh, M. and Najibi, S. M., (2017). Type I general exponential class of distributions. Pakistan Journal of Statistics and Operation Research, Vol. 14, pp. 39–55.

Ibrahim, M., (2019). A new extended Fréchet distribution: properties and estimation. Pakistan Journal of Statistics and Operation Research, Vol. 15, pp. 773–796.

Ibrahim, M., (2020a). The compound Poisson Rayleigh Burr XII distribution: properties and applications. Journal of Applied Probability and Statistics, Vol. 15, pp. 73–97.

Ibrahim, M., (2020b). The generalized odd Log-logistic Nadarajah Haghighi distribution: statistical properties and different methods of estimation. Journal of Applied Probability and Statistics, Vol. 15, pp. 61–84.

Ibrahim, M., Altun, E. and Yousof, H. M., (2020). A new distribution for modeling lifetime data with different methods of estimation and censored regression modeling. Statistics, Optimization & Information Computing, Vol. 8, pp. 610–630.

Ibrahim, M., Yadav, A. S., Yousof, H. M., Goual, H. and Hamedani, G. G., (2019). A new extension of Lindley distribution: modified validation test, characterizations and different methods of estimation, Communications for Statistical Applications and Methods, Vol. 26, pp. 473–495.

Jahanshahi, S. M. A., Yousof, H. M. and Sharma, V. K., (2019). The Burr X Fréchet Model for Extreme Values: Mathematical Properties, Classical Inference and Bayesian Analysis. Pakistan Journal of Statistics and Operation Research, Vol. 15, pp. 797–818.

Johnson, N. L., Kemp, A. W. and Kotz, S., (2005). Univariate discrete distributions, 3rd edn. Wiley, Hoboken.

Korkmaz, M. C., Altun, E., Yousof, H. M. and HAMEDANI G. G., (2019). The odd power Lindley generator of probability distributions: properties, characterizations and regression modeling, International Journal of Statistics and Probability, Vol. 8, pp. 70–89.

Korkmaz, M. C., Yousof, H. M. and Ali, M. M., (2017). Some theoretical and computational aspects of the odd Lindley Fréchet distribution, Journal of Statisticians: Statistics and Actuarial Sciences, Vol. 2, pp. 129–140.

Korkmaz, M. C., Yousof, H. M. and Hamedani G. G., (2018). The exponential Lindley odd log-logistic G family: properties, characterizations and applications. Journal of Statistical Theory and Applications, Vol. 17, pp. 554–571.

Krishna, E., Jose, K. K., Alice, T. and Risti, M. M., (2013). The Marshall-Olkin Fréchet distribution. Communications in Statistics-Theory and Methods, Vol. 42, pp. 4091–4107

Mahmoud, M. R., Mandouh, R. M., (2013). On the transmuted Fréchet distribution. Journal of Applied Sciences Research, Vol. 9, pp. 5553–5561.

Mansour, M. M., Ibrahim, M., Aidi, K., Shafique Butt, N., Ali, M. M., Yousof, H. M. and Hamed, M. S., (2020a). A New Log-Logistic Lifetime Model with Mathematical

Properties, Copula, Modified Goodness-of-Fit Test for Validation and Real Data Modeling. Mathematics, Vol. 8, p. 1508.

Mansour, M. M., Butt, N. S., Ansari, S. I., Yousof, H. M., Ali, M. M. and Ibrahim, M., (2020b). A new exponentiated Weibull distribution’s extension: copula, mathematical properties and applications. Contributions to Mathematics, Vol. 1 (2020), pp. 57–66, doi: 10.47443/cm.2020.0018

Mansour, M., Korkmaz, M. Ç., Ali, M. M., Yousof, H. M., Ansari, S. I. and Ibrahim, M., (2020c). A generalization of the exponentiated Weibull model with properties, Copula and application. Eurasian Bulletin of Mathematics, Vol. 3, pp. 84–102.

Mansour, M., Rasekhi, M., Ibrahim, M., Aidi, K., Yousof, H. M. and Elrazik, E. A., (2020d). A New Parametric Life Distribution with Modified Bagdonavičius– Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods. Entropy, Vol. 22, p. 592.

Mansour, M., Yousof, H. M., Shehata, W. A. and Ibrahim, M., (2020e). A new two parameter Burr XII distribution: properties, copula, different estimation methods and modeling acute bone cancer data. Journal of Nonlinear Science and Applications, Vol.13, pp. 223–238.

Mansour, M. M., Butt, N. S., Yousof, H. M., Ansari, S. I. and Ibrahim, M., (2020f). A Generalization of Reciprocal Exponential Model: Clayton Copula, Statistical Properties and Modeling Skewed and Symmetric Real Data Sets. Pakistan Journal of Statistics and Operation Research, Vol. 16, pp. 373–386.

Merovci, F., Alizadeh, M., Yousof, H. M. and Hamedani G. G., (2017). The exponentiated transmuted-G family of distributions: theory and applications, Communications in Statistics-Theory and Methods, Vol. 46, pp. 10800–10822.

Merovci, F., Yousof, H. M. and Hamedani, G. G., (2020). The Poisson Topp Leone Generator of Distributions for Lifetime Data: Theory, Characterizations and Applications. Pakistan Journal of Statistics and Operation Research, Vol. 16, pp. 343–355.

Nikulin, M. S., (1973a). Chi-square Test for Continuous Distributions with Shift and Scale Parameters. teor. Veroyatn. Primen, Vol. 18, pp. 559–568.

Nikulin, M. S., (1973b). Chi-squared test for continuous distributions with shift and scal parameters. Theory of Probability and its Applications, Vol. 18, pp. 559–568.

Nikulin, M. S., (1973c). On a Chi-squared test for continuous distributions. Theory of Probability and its Aplications, Vol. 19, pp. 638–639.

Salah, M., El-Morshedy, M., Eliwa, M. S. and Yousof, H. M., (2020). Expanded Fréchet Model: Mathematical Properties, Copula, Different Estimation Methods, Applications and Validation Testing. Mathematics, Vol. 8, pp.19–49.

Swain, J. Venkatraman, S. and Wilson, J., (1988). Least squares estimation of distribution function in Johnsons translation system, J. Stat. Comput. Simul., Vol. 29, pp. 271–297.

Nascimento, A. D. C., Silva, K. F., Cordeiro, G. M., Alizadeh, M. and Yousof, H. M., (2019). The odd Nadarajah-Haghighi family of distributions: properties and applications. Studia Scientiarum Mathematicarum Hungarica, Vol. 56, pp. 1–26.

Rao, K. C., Robson, D. S., (1974). A Chi-square statistic for goodness-of-fit tests within the Exponential family. Communication in Statistics, Vol. 3, pp.1139–1153.

Yadav, A. S., Goual, H., Alotaibi, R. M. Rezk, H., Ali, M. M. and Yousof, H. M., (2020). Validation of the Topp-Leone-Lomax model via a modified Nikulin-Rao-Robson goodness-of-fit test with different methods of estimation. Symmetry, Vol. 12, pp. 1–26. DOI: 10.3390/sym12010057

Yousof, H. M., Afify, A. Z., Alizadeh, M., Butt, N. S., Hamedani, G. G. and Ali, M. M., (2015). The transmuted exponentiated generalized-G family of distributions. Pakistan Journal of Statistics and Operation Research, Vol. 11, pp. 441–464.

Yousof, H. M., Afify, A. Z., Ebraheim, A. N., Hamedani, G. G. and Butt, N. S., (2016). On six-parameter Fréchet distribution: properties and applications, Pakistan Journal of Statistics and Operation Research, Vol. 12, pp. 281–299.

Yousof, H. M., Alizadeh, M., Jahanshahiand, S. M. A., Ramires, T. G., Ghosh, I. and Hamedani G. G., (2017). The transmuted Topp-Leone G family of distributions: theory, characterizations and applications, Journal of Data Science. Vol. 15, pp. 723–740.

Yousof, H. M., Afify, A. Z., Alizadeh, M., Nadarajah, S., Aryal, G. R. and Hamedani, G. G., (2018a). The Marshall-Olkin generalized-G family of distributions with Applications, STATISTICA, Vol. 78, pp. 273–295.

Yousof, H. M., Altun, E. and Hamedani, G. G., (2018b). A new extension of Frechet distribution with regression models, residual analysis and characterizations. Journal of Data Science, Vol. 16, pp. 743–770.

Yousof, H. M., Afify, A. Z., Alizadeh, M., Hamedani G. G., Jahanshahi, S. M. A. and Ghosh, I., (2018c). The generalized transmuted Poisson-G family of Distributions. Pakistan Journal of Statistics and Operation Research, Vol. 14, pp. 759–779.

Yousof, H. M., Altun, E., Ramires, T. G., Alizadeh, M. and Rasekhi, M., (2018c). A new family of distributions with properties, regression models and applications, Journal of Statistics and Management Systems, Vol. 21, pp. 163–188.

Yousof, H. M., Butt, N. S., Alotaibi, R., Rezk, H. R., Alomani, G. A. and Ibrahim, M., (2019). A new compound Fréchet distribution for modeling breaking stress and strengths data. Pakistan Journal of Statistics and Operation Research, Vol. 15, pp. 1017–1035.

Yousof, H. M., Hamedani, G. G. and Ibrahim, M., (2020). The Two-parameter Xgamma Fréchet Distribution: Characterizations, Copulas, Mathematical Properties and Different Classical Estimation Methods. Contributions to Mathematics, Vol. 2 (2020), pp. 32–41.

Yousof, H. M., Rasekhi, M., Altun, E. and Alizadeh, M., (2018d). The extended odd Frechet family of distributions: properties, applications and regression modeling. International Journal of Applied Mathematics and Statistics, Vol. 30, pp. 1–30.