Cubic rank transmuted inverse rayleigh distribution Properties and applications

Caner TANIŞ; Buğra SARAÇOĞLU

doi:10.14744/sigma.2022.00042

Article Open Access1 January 2022

Cubic rank transmuted inverse rayleigh distribution Properties and applications

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Caner TANIŞ ¹

¹Çankırı Karatekin University

Sigma Journal of Engineering and Natural Sciences 2022, Vol. 40, Issue 2, pp. 421-432; doi.org/10.14744/sigma.2022.00042

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Abstract

In this paper, we propose a new lifetime distribution called Cubic Rank Transmuted Inverse Rayleigh as an alternative to the inverse Rayleigh distribution. Some distributional properties of the suggested distribution such as moments, incomplete moments, Bonferroni and Lorenz curves, moment generating function, quantile function, median, mean residual life function are examined. We consider five methods such as maximum likelihood, the least squares, weighted least squares, Anderson Darling method, and Crámer–von-Mises method to estimate the parameters of the proposed distribution. Furthermore, a comprehensive Monte Carlo simulation study is performed to compare the performances of the examined estimators according to mean square errors and biases. Finally, a real data application is given to illustrate the usefulness of the proposed distribution.

Keywords: Cubic Rank Transmuted Inverse Rayleigh Distribution; Point Estimation; Bonferroni Curve; Lorenz Curve; Monte Carlo Simulation

References

illustrates that the best-fitted model is CRTIR (α,λ1,λ2) distribution among all fitted distributions in modeling [1] Shaw WT, Buckley IR. The alchemy of probability precipitation data. distributions: Beyond gram charlier & cornish fisher expansions, and skew-normalor kurtotic-normal distributions. 2007. Available at: http://citeseerx.ist. AUTHORSHIP CONTRIBUTIONS psu.edu/viewdoc/download?doi=10.1.1.335.4244&r Authors equally contributed to this work. ep=rep1&type=pdf. Accessed May 26, 2022. 432 Sigma J Eng Nat Sci, Vol. 40, No. 2, pp. 421–432, June, 2022
Shaw WT, Buckley IR. The alchemy of probability [13] Granzotto DCT, Louzada F, Balakrishnan N. distributions: Beyond Gram-Charlier expansions, Cubic rank transmuted distributions: Inferential and a skew-kurtotic-normal distribution from issues and applications. J Statistical Comput Simul a rank transmutation map. arXiv preprint arXiv 2017;87:2760–2778. [CrosssRef] 2009. Available at: https://arxiv.org/abs/0901.0434. [14] Aslam M, Hussain Z, Asghar Z. Cubic transmuted-g Accessed May 26, 2022. family of distributions and its properties. Stoch Qual
Ahmad A, Ahmad SP, Ahmed A. Transmuted Control 2018;33:103–112. [CrosssRef] inverse Rayleigh distribution: A generalization of [15] Saraçoğlu B, Tanış C. A new statistical distribution: the inverse Rayleigh distribution. Math Theory Cubic rank transmuted Kumaraswamy distribution Model 2014;4:90–98. and its properties. J Nat Sci Foundation Sri Lanka
Granzotto DCT, Louzada F. The transmuted log- 2018;46:505–518. [CrosssRef] logistic distribution: Modeling, inference, and an [16] Bhatti FA, Hamedani GG, Najibi SM, Ahmad M. application to a polled tabapua race time up to Cubic rank transmuted modified burr III distri- first calving data. Commun Stat Theory Methods bution: Development, properties, characteriza- 2015;44:3387–3402. [CrosssRef] tions and applications. Data Sci J 2020;18:299–318. [CrosssRef]
Alizadeh M, Merovci F, Hamedani GG. Generalized transmuted family of distributions: Properties and [17] Bhatti FA, Hamedani GG, Sheng W, Ahmad M. applications. Hacettepe J Math Stat 2017;46:645– Cubic rank transmuted modified burr III pareto
Merovci F, Alizadeh M, Hamedani GG. Another tions and applications. Int J Stat Probab 2019;8:94– generalized transmuted family of distribu- 112. [CrosssRef] tions: Properties and applications. Austrian J Stat [18] Hameldarbandı M, Yılmaz M. Some comments on 2016;45:71–93. [CrosssRef] methodology of cubic rank transmuted distribu-
Nofal ZM, Afify AZ, Yousof HM, Granzotto DC, tions. Commun Fac Sci Univ Ankara Series A1 Louzada F. Kumaraswamy transmuted exponenti- Math Stat 2020;69:167–176. [CrosssRef] ated additive Weibull distribution. Int J Stat Probab [19] Bonferroni CE. Elementi di statistica generale. 2016;5:78–99. [CrosssRef] Firenze: Libreria Seeber; 1930.
Merovci F, Alizadeh M, Yousof HM, Hamedani GG. [20] Swain JJ, Venkatraman S, Wilson J. Least squares esti- The exponentiated transmuted-G family of distri- mation of distribution function in Johnson’s transla- butions: Theory and applications. Commun Stat tion system. J Stat Comput Simul 1988;29:271–297. [CrosssRef] Theory Methods 2017;46:10800–10822. [CrosssRef] [21] Shahbaz MQ, Shahbaz S, Butt NS. The kumaras-
Bhatti FA, Hamedani GG, Korkmaz MÇ, Ahmad wamy-inverse weibull distribution. Pakistan J Stat M. The transmuted geometric-quadratic hazard rate Oper Res 2012;8:479–489. [CrosssRef] distribution: Development, properties, characteriza- [22] Hinkley D. On quick choice of power transforma- tions and applications. J Stat Distrib Appl 2018;5:1– tions. J R Stat Soc Ser C Appl Stat 1977;26:67–69.
Alizadeh M, Yousof HM, Afify AZ, Cordeiro GM, [23] Haq MA. Transmuted exponentiated inverse Mansoor M. The complementary generalized trans- Rayleigh distribution. J Stat Appl Prob 2016;5:337– muted Poisson-G family of distributions. Austrian J 433. [CrosssRef] Stat 2018;47:60–80. [CrosssRef] [24] Usman RM, Haq M, Talib J. Kumaraswamy half-
Tanış C, Saraçoğlu B, Kuş C, Pekgör A. Transmuted logistic distribution: properties and applications. J complementary exponential power distribution: Stat Appl Probab 2017;6:597–609. [CrosssRef] Properties and applications. Cumhuriyet Sci J [25] Elgarhy M, Elbatal I, ul Haq MA, Hassan AS. 2020;41:419–432. [CrosssRef] Transmuted Kumaraswamy quasi Lindley distribu-
Saraçoğlu B, Tanış C. A new lifetime distribu- tion with applications. Annals Data Sci 2018;5:565– tion: Transmuted exponential power distribution. 581. [CrosssRef] Commun Fac Sci Univ Ankara Series A1 Math Stat 2021;70:1–14. [CrosssRef]

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TANIŞ, C.; SARAÇOĞLU, B. Cubic rank transmuted inverse rayleigh distribution Properties and applications. Sigma Journal of Engineering and Natural Sciences 2022, Vol. 40, pp. 421-432. https://doi.org/10.14744/sigma.2022.00042

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Cubic rank transmuted inverse rayleigh distribution Properties and applications

¹Çankırı Karatekin University

Abstract

References

illustrates that the best-fitted model is CRTIR (α,λ1,λ2) distribution among all fitted distributions in modeling [1] Shaw WT, Buckley IR. The alchemy of probability precipitation data. distributions: Beyond gram charlier & cornish fisher expansions, and skew-normalor kurtotic-normal distributions. 2007. Available at: http://citeseerx.ist. AUTHORSHIP CONTRIBUTIONS psu.edu/viewdoc/download?doi=10.1.1.335.4244&r Authors equally contributed to this work. ep=rep1&type=pdf. Accessed May 26, 2022. 432 Sigma J Eng Nat Sci, Vol. 40, No. 2, pp. 421–432, June, 2022
Shaw WT, Buckley IR. The alchemy of probability [13] Granzotto DCT, Louzada F, Balakrishnan N. distributions: Beyond Gram-Charlier expansions, Cubic rank transmuted distributions: Inferential and a skew-kurtotic-normal distribution from issues and applications. J Statistical Comput Simul a rank transmutation map. arXiv preprint arXiv 2017;87:2760–2778. [CrosssRef] 2009. Available at: https://arxiv.org/abs/0901.0434. [14] Aslam M, Hussain Z, Asghar Z. Cubic transmuted-g Accessed May 26, 2022. family of distributions and its properties. Stoch Qual
Ahmad A, Ahmad SP, Ahmed A. Transmuted Control 2018;33:103–112. [CrosssRef] inverse Rayleigh distribution: A generalization of [15] Saraçoğlu B, Tanış C. A new statistical distribution: the inverse Rayleigh distribution. Math Theory Cubic rank transmuted Kumaraswamy distribution Model 2014;4:90–98. and its properties. J Nat Sci Foundation Sri Lanka
Granzotto DCT, Louzada F. The transmuted log- 2018;46:505–518. [CrosssRef] logistic distribution: Modeling, inference, and an [16] Bhatti FA, Hamedani GG, Najibi SM, Ahmad M. application to a polled tabapua race time up to Cubic rank transmuted modified burr III distri- first calving data. Commun Stat Theory Methods bution: Development, properties, characteriza- 2015;44:3387–3402. [CrosssRef] tions and applications. Data Sci J 2020;18:299–318. [CrosssRef]
Alizadeh M, Merovci F, Hamedani GG. Generalized transmuted family of distributions: Properties and [17] Bhatti FA, Hamedani GG, Sheng W, Ahmad M. applications. Hacettepe J Math Stat 2017;46:645– Cubic rank transmuted modified burr III pareto
Merovci F, Alizadeh M, Hamedani GG. Another tions and applications. Int J Stat Probab 2019;8:94– generalized transmuted family of distribu- 112. [CrosssRef] tions: Properties and applications. Austrian J Stat [18] Hameldarbandı M, Yılmaz M. Some comments on 2016;45:71–93. [CrosssRef] methodology of cubic rank transmuted distribu-
Nofal ZM, Afify AZ, Yousof HM, Granzotto DC, tions. Commun Fac Sci Univ Ankara Series A1 Louzada F. Kumaraswamy transmuted exponenti- Math Stat 2020;69:167–176. [CrosssRef] ated additive Weibull distribution. Int J Stat Probab [19] Bonferroni CE. Elementi di statistica generale. 2016;5:78–99. [CrosssRef] Firenze: Libreria Seeber; 1930.
Merovci F, Alizadeh M, Yousof HM, Hamedani GG. [20] Swain JJ, Venkatraman S, Wilson J. Least squares esti- The exponentiated transmuted-G family of distri- mation of distribution function in Johnson’s transla- butions: Theory and applications. Commun Stat tion system. J Stat Comput Simul 1988;29:271–297. [CrosssRef] Theory Methods 2017;46:10800–10822. [CrosssRef] [21] Shahbaz MQ, Shahbaz S, Butt NS. The kumaras-
Bhatti FA, Hamedani GG, Korkmaz MÇ, Ahmad wamy-inverse weibull distribution. Pakistan J Stat M. The transmuted geometric-quadratic hazard rate Oper Res 2012;8:479–489. [CrosssRef] distribution: Development, properties, characteriza- [22] Hinkley D. On quick choice of power transforma- tions and applications. J Stat Distrib Appl 2018;5:1– tions. J R Stat Soc Ser C Appl Stat 1977;26:67–69.
Alizadeh M, Yousof HM, Afify AZ, Cordeiro GM, [23] Haq MA. Transmuted exponentiated inverse Mansoor M. The complementary generalized trans- Rayleigh distribution. J Stat Appl Prob 2016;5:337– muted Poisson-G family of distributions. Austrian J 433. [CrosssRef] Stat 2018;47:60–80. [CrosssRef] [24] Usman RM, Haq M, Talib J. Kumaraswamy half-
Tanış C, Saraçoğlu B, Kuş C, Pekgör A. Transmuted logistic distribution: properties and applications. J complementary exponential power distribution: Stat Appl Probab 2017;6:597–609. [CrosssRef] Properties and applications. Cumhuriyet Sci J [25] Elgarhy M, Elbatal I, ul Haq MA, Hassan AS. 2020;41:419–432. [CrosssRef] Transmuted Kumaraswamy quasi Lindley distribu-
Saraçoğlu B, Tanış C. A new lifetime distribu- tion with applications. Annals Data Sci 2018;5:565– tion: Transmuted exponential power distribution. 581. [CrosssRef] Commun Fac Sci Univ Ankara Series A1 Math Stat 2021;70:1–14. [CrosssRef]

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[1] illustrates that the best-fitted model is CRTIR (α,λ1,λ2) distribution among all fitted distributions in modeling [1] Shaw WT, Buckley IR. The alchemy of probability precipitation data. distributions: Beyond gram charlier & cornish fisher expansions, and skew-normalor kurtotic-normal distributions. 2007. Available at: http://citeseerx.ist. AUTHORSHIP CONTRIBUTIONS psu.edu/viewdoc/download?doi=10.1.1.335.4244&r Authors equally contributed to this work. ep=rep1&type=pdf. Accessed May 26, 2022. 432 Sigma J Eng Nat Sci, Vol. 40, No. 2, pp. 421–432, June, 2022

[2] Shaw WT, Buckley IR. The alchemy of probability [13] Granzotto DCT, Louzada F, Balakrishnan N. distributions: Beyond Gram-Charlier expansions, Cubic rank transmuted distributions: Inferential and a skew-kurtotic-normal distribution from issues and applications. J Statistical Comput Simul a rank transmutation map. arXiv preprint arXiv 2017;87:2760–2778. [CrosssRef] 2009. Available at: https://arxiv.org/abs/0901.0434. [14] Aslam M, Hussain Z, Asghar Z. Cubic transmuted-g Accessed May 26, 2022. family of distributions and its properties. Stoch Qual

[3] Ahmad A, Ahmad SP, Ahmed A. Transmuted Control 2018;33:103–112. [CrosssRef] inverse Rayleigh distribution: A generalization of [15] Saraçoğlu B, Tanış C. A new statistical distribution: the inverse Rayleigh distribution. Math Theory Cubic rank transmuted Kumaraswamy distribution Model 2014;4:90–98. and its properties. J Nat Sci Foundation Sri Lanka

[4] Granzotto DCT, Louzada F. The transmuted log- 2018;46:505–518. [CrosssRef] logistic distribution: Modeling, inference, and an [16] Bhatti FA, Hamedani GG, Najibi SM, Ahmad M. application to a polled tabapua race time up to Cubic rank transmuted modified burr III distri- first calving data. Commun Stat Theory Methods bution: Development, properties, characteriza- 2015;44:3387–3402. [CrosssRef] tions and applications. Data Sci J 2020;18:299–318. [CrosssRef]

[5] Alizadeh M, Merovci F, Hamedani GG. Generalized transmuted family of distributions: Properties and [17] Bhatti FA, Hamedani GG, Sheng W, Ahmad M. applications. Hacettepe J Math Stat 2017;46:645– Cubic rank transmuted modified burr III pareto

[6] Merovci F, Alizadeh M, Hamedani GG. Another tions and applications. Int J Stat Probab 2019;8:94– generalized transmuted family of distribu- 112. [CrosssRef] tions: Properties and applications. Austrian J Stat [18] Hameldarbandı M, Yılmaz M. Some comments on 2016;45:71–93. [CrosssRef] methodology of cubic rank transmuted distribu-

[7] Nofal ZM, Afify AZ, Yousof HM, Granzotto DC, tions. Commun Fac Sci Univ Ankara Series A1 Louzada F. Kumaraswamy transmuted exponenti- Math Stat 2020;69:167–176. [CrosssRef] ated additive Weibull distribution. Int J Stat Probab [19] Bonferroni CE. Elementi di statistica generale. 2016;5:78–99. [CrosssRef] Firenze: Libreria Seeber; 1930.

[8] Merovci F, Alizadeh M, Yousof HM, Hamedani GG. [20] Swain JJ, Venkatraman S, Wilson J. Least squares esti- The exponentiated transmuted-G family of distri- mation of distribution function in Johnson’s transla- butions: Theory and applications. Commun Stat tion system. J Stat Comput Simul 1988;29:271–297. [CrosssRef] Theory Methods 2017;46:10800–10822. [CrosssRef] [21] Shahbaz MQ, Shahbaz S, Butt NS. The kumaras-

[9] Bhatti FA, Hamedani GG, Korkmaz MÇ, Ahmad wamy-inverse weibull distribution. Pakistan J Stat M. The transmuted geometric-quadratic hazard rate Oper Res 2012;8:479–489. [CrosssRef] distribution: Development, properties, characteriza- [22] Hinkley D. On quick choice of power transforma- tions and applications. J Stat Distrib Appl 2018;5:1– tions. J R Stat Soc Ser C Appl Stat 1977;26:67–69.

[10] Alizadeh M, Yousof HM, Afify AZ, Cordeiro GM, [23] Haq MA. Transmuted exponentiated inverse Mansoor M. The complementary generalized trans- Rayleigh distribution. J Stat Appl Prob 2016;5:337– muted Poisson-G family of distributions. Austrian J 433. [CrosssRef] Stat 2018;47:60–80. [CrosssRef] [24] Usman RM, Haq M, Talib J. Kumaraswamy half-

[11] Tanış C, Saraçoğlu B, Kuş C, Pekgör A. Transmuted logistic distribution: properties and applications. J complementary exponential power distribution: Stat Appl Probab 2017;6:597–609. [CrosssRef] Properties and applications. Cumhuriyet Sci J [25] Elgarhy M, Elbatal I, ul Haq MA, Hassan AS. 2020;41:419–432. [CrosssRef] Transmuted Kumaraswamy quasi Lindley distribu-

[12] Saraçoğlu B, Tanış C. A new lifetime distribu- tion with applications. Annals Data Sci 2018;5:565– tion: Transmuted exponential power distribution. 581. [CrosssRef] Commun Fac Sci Univ Ankara Series A1 Math Stat 2021;70:1–14. [CrosssRef]

Published	1 January 2022
Version	v1
Access	Open Access

Published	1 January 2022
Version	v1
Access	Open Access

Cubic rank transmuted inverse rayleigh distribution Properties and applications

Abstract

References

Share and Cite

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Cubic rank transmuted inverse rayleigh distribution Properties and applications

Abstract

References

Share and Cite

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