# COMMENTS ON A ZUBAIR–G FAMILY OF CUMULATIVE LIFETIME DISTRIBUTIONS. SOME EXTENSIONS

 Title COMMENTS ON A ZUBAIR–G FAMILY OF CUMULATIVE LIFETIME DISTRIBUTIONS. SOME EXTENSIONS Publication Type Journal Article Year of Publication 2019 Authors KYURKCHIEV, NIKOLAY, ILIEV, ANTON, RAHNEV, ASEN Volume 23 Issue 1 Start Page 1 Pagination 20 Date Published 11/2018 ISSN 1083-2564 AMS 41A46, 68N30 Abstract In this paper we study the one--sided Hausdorff approximation of the shifted Heaviside step function by a class of the Zubair--G family of cumulative lifetime distribution. The estimates of the value of the best Hausdorff approximation obtained in this article can be used in practice as one possible additional criterion in ''saturation'' study. As an illustrative example we consider the modelling of the growth of red abalone (Haliotis Rufescens) in Northern California. We also look at a possible extension, which we call $\alpha$--Zubair--G Family. For the analysis of said dataset with the new (cdf), some comparisons are made. Finally, the potentiality of the software reliability models analyzed bu means of real dataset. Numerical examples, illustrating our results are presented using programming environment CAS Mathematica. URL https://acadsol.eu/en/articles/23/1/1.pdf DOI 10.12732/caa.v23i1.1 Refereed Designation Refereed Full Text [1] Z. Ahmad, The Zubair-G Family of Distributions: Properties and Applications, Annals of Data Science, (2018), doi: 10.1007/s40745-018-0169-9. [2] Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Stat. and Prob. Letters, 49 (2000), 155-161. [3] M. Xie, Y. Tang, T. Goh, A modified Weibull extension with bathtubZUBAIR-G FAMILY OF CUMULATIVE LIFETIME DISTRIBUTIONS 15 shaped failure rate function, Reliability Eng. and System Safety, 76 (2002), 279-285. [4] M. Khan, A. Sharma, Generalized order statistics from Chen distribution and its characterization, J. of Stat. Appl. and Prob., 5 (2016), 123-128. [5] S. Dey, D. Kumar, P. Ramos, F. Louzada, Exponentiated Chen distribution: Properties and Estimations, Comm. in Stat.-Simulation and Computation, 46, No. 10 (2017), 8118-8139. [6] Y. Chaubey, R. Zhang, An extension of Chen’s family of survival distributions with bathtub shape or increasing hazard rate function, Comm. in Stat.-Theory and Methods, 44 (2015), 4049-4069. [7] D. Kumar, M. Kumar, A new generalization of the extended exponential distribution with application, Ann. Data Sci. (2018), 22 pp. [8] M. Khan, Transmuted generalized power Weibull distribution, Thailand Statistician, 16, No. 2 (2018), 156-172. [9] G. Cordeiro, A. Afify, E. Ortega, A. Suzuki, M. Mead, The odd Lomax generator of distributions, J. Comp. Appl. Math., 377 (2019), 222-237. [10] V. Cancho, F. Louzada, G. Barriga, The Poisson-exponential lifetime distribution, Comp. Stat. Data Anal., 55 (2011), 677-686. [11] G. Rodrigues, F. Louzada, P. Ramos, Poisson-exponential distribution: different methods of estimation, J. of Appl. Stat., 45, No. 1 (2018), 128144. [12] F. Louzada, P. Ramos, P. Ferreira, Exponential-Poisson distribution: estimation and applications to rainfall and aircraft data with zero occurrence, Communication in Statistics-Simulation and Computation, (2018), doi: 10.1080/03610918.2018.1491988. [13] P. Ramos, D. Dey, F. Louzada, V. Lachos, An extended Poisson family of lifetime distribution: A unified approach in competitive and Complementary risk, arXiv: submit/2267507 [stat.AP], (2018). 16 N. KYURKCHIEV, A. ILIEV, AND A. RAHNEV [14] N. Kyurkchiev, S. Markov, On the Hausdorff distance between the Heaviside step function and Verhulst logistic function, J. Math. Chem., 54, No. 1 (2016), 109-119. [15] N. Kyurkchiev, S. Markov, Sigmoid functions: Some Approximation and Modelling Aspects, LAP LAMBERT Academic Publishing, Saarbrucken (2015), ISBN 978-3-659-76045-7. [16] N. Kyurkchiev, A. Iliev, S. Markov, Some Techniques for Recurrence Generating of Activation Functions: Some Modeling and Approximation Aspects, LAP LAMBERT Academic Publishing (2017), ISBN: 978-3-33033143-3. [17] R. Anguelov, M. Borisov, A. Iliev, N. Kyurkchiev, S. Markov, On the chemical meaning of some growth models possessing Gompertzian-type property, Math. Meth. Appl. Sci., (2017), 1-12, doi: 10.1002/mma.4539. [18] R. Anguelov, N. Kyurkchiev, S. Markov, Some properties of the Blumberg’s hyper-log-logistic curve, BIOMATH, 7, No. 1 (2018), 8 pp. [19] A. Iliev, N. Kyurkchiev, S. Markov, On the Approximation of the step function by some sigmoid functions, Mathematics and Computers in Simulation, 133 (2017), 223-234. [20] A. Iliev, N. Kyurkchiev, S. Markov, Approximation of the cut function by Stannard and Richards sigmoid functions, IJPAM, 109, No. 1 (2016), 119-128. [21] S. Markov, A. Iliev, A. Rahnev, N. Kyurkchiev, A note on the Loglogistic and transmuted Log-logistic models. Some applications, Dynamic Systems and Applications, 27, No. 3 (2018), 593-607. [22] S. Markov, N. Kyurkchiev, A. Iliev, A. Rahnev, On the approximation of the cut functions by hyper-log-logistic function, Neural, Parallel and Scientific Computations, 26, No. 2 (2018), 169-182. [23] N. Kyurkchiev, A. Iliev, S. Markov, Families of recurrence generated three and four parametric activation functions, Int. J. Sci. Res. and Development, 4, No. 12 (2017), 746-750. ZUBAIR-G FAMILY OF CUMULATIVE LIFETIME DISTRIBUTIONS 17 [24] N. Kyurkchiev, A note on the new geometric representation for the parameters in the fibril elongation process, C. R. Acad. Bulg. Sci., 69, No. 8 (2016), 963-972. [25] N. Kyurkchiev, On the numerical solution of the general ”ligand-gated neuroreceptors model’ via CAS Mathematica, Pliska Stud. Math. Bulgar., 26 (2016), 133-142. [26] N. Kyurkchiev, S. Markov, On the numerical solution of the general kinetic ”K-angle” reaction system, Journal of Mathematical Chemistry, 54, No. 3 (2016), 792-805. [27] S. Markov, N. Kyurkchiev, A. Iliev, A. Rahnev, On the approximation of the generalized cut functions of degree p+1 by smooth hyper-log-logistic function, Dynamic Systems and Applications, 27, No. 4 (2018), 715-728. [28] S. Markov, A. Iliev, A. Rahnev, N. Kyurkchiev, A Note On the Threestage Growth Model, Dynamic Systems and Applications, 28, No. 1 (2019), 63-72. [29] S. Markov, A. Iliev, A. Rahnev, N. Kyurkchiev, A Note On the n-stage Growth Model. Overview, Biomath Communications, 5, No. 2 (2018). (accepted) [30] O. Rahneva, H. Kiskinov, I. Dimitrov, V. Matanski, Application of a Weibull Cumulative Distribution Function Based on m Existing Ones to Population Dynamics, International Electronic Journal of Pure and Applied Mathematics, 12, No. 1 (2018), 111-121. [31] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Some software reliability models: Approximation and modeling aspects, LAP LAMBERT Academic Publishing (2018), ISBN: 978-613-9-82805-0. [32] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Nontrivial Models in Debugging Theory (Part 2), LAP LAMBERT Academic Publishing (2018), ISBN: 978-613-9-87794-2. [33] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, On the extended Chen’s and Pham’s software reliability models. Some applications, Int. J. of Pure and Appl. Math., 118, No. 4 (2018), 1053-1067. 18 N. KYURKCHIEV, A. ILIEV, AND A. RAHNEV [34] N. Pavlov, G. Spasov, A. Rahnev, N. Kyurkchiev, A new class of Gompertz-type software reliability models, International Electronic Journal of Pure and Applied Mathematics, 12, No. 1 (2018), 43-57. [35] N. Pavlov, G. Spasov, A. Rahnev, N. Kyurkchiev, Some deterministic reliability growth curves for software error detection: Approximation and modeling aspects, International Journal of Pure and Applied Mathematics, 118, No. 3 (2018), 599-611. [36] N. Pavlov, A. Golev, A. Rahnev, N. Kyurkchiev, A note on the Yamadaexponential software reliability model, International Journal of Pure and Applied Mathematics, 118, No. 4 (2018), 871-882. [37] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, A Note on The ”Mean Value” Software Reliability Model, International Journal of Pure and Applied Mathematics, 118, No. 4 (2018), 949-956. [38] D. R. Jeske, X. Zhang, Some successful approaches to software reliability modeling in industry, J. Syst. Softw., 74 (2005), 85-99. [39] K. Song, H. Pham, A Software Reliability Model with a Weibull Fault Detection Rate Function Subject to Operating Environments, Appl. Sci., 7 (2017), 16 pp., doi: 10.3390/app7100983. [40] N. Pavlov, A. Golev, A. Rahnev, N. Kyurkchiev, A note on the generalized inverted exponential software reliability model, International Journal of Advanced Research in Computer and Communication Engineering, 7, No. 3 (2018), 484-487. [41] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Transmuted inverse exponential software reliability model, Int. J. of Latest Research in Engineering and Technology, 4, No. 5 (2018), 1-6. [42] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Analysis of the Chen’s and Pham’s Software Reliability Models, Cybernetics and Information Technologies, 18, No. 3 (2018), 37-47. [43] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, On Some Nonstandard Software Reliability Models, Dynamic Systems and Applications, 27, No. 4 (2018), 757-771. ZUBAIR-G FAMILY OF CUMULATIVE LIFETIME DISTRIBUTIONS 19 [44] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Some deterministic growth curves with applications to software reliability analysis, Int. J. of Pure and Appl. Math., 119, No. 2 (2018), 357-368. [45] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Investigations of the Kstage Erlangian software reliability growth model, Int. J. of Pure and Appl. Math., 119, No. 3 (2018), 441-449. [46] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Some Transmuted Software Reliability Models, Journal of Mathematical Sciences and Modelling, 1, No. 2 (2018). (to appear) [47] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, A Note on Ohbas Inflexion S-shaped Software Reliability Growth Model, Collection of scientific works from conference Mathematics. Informatics. Information Technologies. Application in Education, Pamporovo, Bulgaria, October 10-12, (2018). (to appear) [48] V. Kyurkchiev, A. Malinova, O. Rahneva, P. Kyurkchiev, On the Burr XII-Weibull Software Reliability Model, Int. J. of Pure and Appl. Math., 119, No. 4 (2018), 639-650. [49] V. Kyurkchiev, A. Malinova, O. Rahneva, P. Kyurkchiev, Some Notes on the Extended Burr XII Software Reliability Model, Int. J. of Pure and Appl. Math., 120 No. 1 (2018), 127-136. [50] N. Kyurkchiev, A. Iliev, Extension of Gompertz-type Equation in Modern Science: 240 Anniversary of the birth of B. Gompertz, LAP LAMBERT Academic Publishing (2018), ISBN: 978-613-9-90569-0. [51] F. Hausdorff, Set Theory (2 ed.) (Chelsea Publ., New York, (1962 [1957]) (Republished by AMS-Chelsea 2005), ISBN: 978-0-821-83835-8. [52] R. Anguelov, S. Markov, Hausdorff Continuous Interval Functions and Approximations, In: SCAN 2014 Proceedings, LNCS, ed. by J.W.von Gudenberg, Springer, Berlin (2015). [53] R. Anguelov, S. Markov, B. Sendov, On the Normed Linear Space of Hausdorff Continuous Functions. In: Lirkov, I., et al. (Eds.): Lecture Notes in Computer Science 3743, Springer (2006), 281-288. 20 N. KYURKCHIEV, A. ILIEV, AND A. RAHNEV [54] R. Anguelov, S. Markov, B. Sendov, Algebraic Operations on the Space of Hausdorff Continuous Functions. In: Bojanov, B. (Ed.): Constructive Theory of Functions, Prof. M. Drinov Academic Publ. House, Sofia (2006), 35-44. [55] R. Anguelov, S. Markov, B. Sendov, The Set of Hausdorff Continuous Functions - the Largest Linear Space of Interval Functions, Reliable Computing, 12 (2006), 337-363. [56] L. Rogers-Bennett, D. W. Rogers, S. A. Schultz, Modeling growth and mortality of red abalone Haliotis Rufescens in Northern California, J. of Shellfish Research, 26, No. 3 (2007), 719-727. [57] H. Pham, System Software Reliability, In: Springer Series in Reliability Engineering, Springer-Verlag London Limited (2006). [58] M. Ohba, Software reliability analysis models, IBM J. Research and Development, 21, No. 4 (1984).