A new alpha power transformed family of distributions: properties and applications to the Weibull model
- Department of Mathematics and Statistics, College of Science, Al Imam Mohammad Ibn Saud Islamic University, (IMSIU), Saudi Arabia.
- Department of Statistics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan.
- Vice Presidency for Graduate Studies and Scientific Research, University of Jeddah, Jeddah, KSA.
Abdullah M. Almarashi
- Statistics Department, Faculty of Science, King AbdulAziz University, Jeddah, Kingdom of Saudi Arabia.
In this article, a new technique of alpha-power transformation is used to propose a new class of lifetime distributions. Four special models of the new family are presented. Some mathematical properties of the proposed model including estimation of the unknown parameters using the method of maximum likelihood are discussed. For the illustrative purposes of the new proposal, a three-parameter special model of this class, namely, new alpha-power transformed Weibull distribution is considered in detail. The proposed distribution offers greater distributional flexibility and is able to model data with increasing, decreasing, and constant or more importantly with bathtub-shaped failure rates. Type-1 and Type-II censoring estimation are discussed. A simulation study based on complete sample of the new model is also carried out. Finally, the usefulness and efficiency of the new proposal is illustrated by analyzing two real data sets.
- Alpha- power transformation
- Weibull distribution
- type-I and type-II censoring
- bathtub shape
- moment generating function
- maximum likelihood estimation
Z. Ahmad, Z. Hussain , Flexible Weibull Extended Distribution, MAYFEB J. Materials Sci., 2 (2017), 5–18.
A. M. Almarashi, M. Elgarhy, A new muth generated family of distributions with applications, J. Nonlinear Sci. Appl., 11 (2018), 1171–1184.
M. Bourguignon, R. B. Silva, G. M. Cordeiro, The Weibull–G family of probability distributions, J. Data Sci., 12 (2014), 53–68. 1
I. W. Burr , Cumulative frequency functions, Ann. Math. Statistics, 13 (1942), 215–232.
G. M. Cordeiro, M. de Castro , A new family of generalized distributions, J. Stat. Comput. Simul., 81 (2011), 883–893.
G. M. Cordeiro, E. M. M. Ortega, S. Nadarajah , The Kumaraswamy Weibull distribution with application to failure data , J. Franklin Inst., 347 (2010), 1399–1429.
S. Dey, V. K. Sharma, M. Mesfioui, A New Extension of Weibull Distribution with Application to Lifetime Data, Annals Data Sci., 4 (2017), 31–61.
M. Elgarhy, M. Haq, G. Ozel, N. Arslan, A new exponentiated extended family of distributions with Applications, Gazi University J. Sci., 30 (2017), 101–115.
M. Elgarhy, A. S. Hassan, M. Rashed, Garhy-generated family of distributions with application, Math. Theory Model., 6 (2016), 1–15. 1
N. Eugene, C. Lee, F. Famoye, The beta-normal distribution and its applications, Comm. Statist. Theory Methods, 31 (2002), 497–512. 1
M. Haq, M. Elgarhy, The odd Frchet-G family of probability distributions, J. Stat. Appl. Prob., 7 (2018), 185–201.
A. S. Hassan, M. Elgarhy , A New family of exponentiated Weibull-generated distributions, Int. J. Math. Appl., 4 (2016), 135–148.
A. S. Hassan, M. Elgarhy, Kumaraswamy Weibull-generated family of distributions with applications, Adv. Appl. Stat., 48 (2016), 205–239. 1
A. S. Hassan, M. Elgarhy, M. Shakil, Type II half Logistic family of distributions with applications, Pak. J. Stat. Oper. Res., 13 (2017), 245–264.
S. Kotz, D. Vicari , Survey of developments in the theory of continuous skewed distributions, Metron, 63 (2005), 225–261.
C. D. Lai, M. Xie, D. N. P. Murthy , A modified Weibull distribution , IEEE Trans. Reliab., 52 (2003), 33–47.
A. Mahdavi, D. Kundu, A new method for generating distributions with an application to exponential distribution, Comm. Statist. Theory Methods, 46 (2017), 6543–6557.
A. W. Marshall, I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, 84 (1997), 641–652.
G. S. Mudholkar, D. K. Srivastava , Exponentiated Weibull family for analyzing bathtub failure rate data, IEEE Trans. Reliab., 42 (1993), 299–302. 1, 1
S. Nadarajah, S. Kotz, Strength modeling using Weibull distributions, J. Mech. Sci. Tech., 22 (2008), 1247–1254.
G. P. Patil, J. K. Ord, On size-biased sampling and related form-invariant weighted distributions, Sankhya, Ser. B, 38 (1976), 48–61.
G. P. Patil, C. R. Rao, The weighted distributions: a survey of their applications, In Applications of Statistics, P. R. Krishnaiah (ed.), 383–405, North Holland Publishing Company, Amesterdam (1977)
K. Pearson, Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material, Philos. Trans. R. Soc. Lond. A, 186 (1895), 343–414.
A. Saboor, T. K. Pogany, Marshall–Olkin gamma–Weibull distribution with applications, Comm. Statist. Theory Methods, 45 (2016), 1550–1563.