The Performance of Estimators for Generalization of Crack Distribution

ec86aa14-6f2c-40fc-8811-658e364db17c20210402045550515wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40513220213220212010.37394/23206.2021.20https://wseas.org/wseas/cms.action?id=23278The Performance of Estimators for Generalization of Crack DistributionSupitchaMamuangbonDepartment of Mathematics and Statistics, Thammasat University, Pathum Thani, ThailandKamonBudsabaDepartment of Mathematics and Statistics, Thammasat University, Pathum Thani, ThailandAndreiVolodinDepartment of Mathematics and Statistics, University of Regina, Saskatchewan, CanadaIn this research, we propose a new four parameter family of distributions called Generalized Crack distribution. We generalizes the family three parameter Crack distribution. The Generalized Crack distribution is a mixture of two parameter Inverse Gaussian distribution, Length-Biased Inverse Gaussian distribution, Twice Length-Biased Inverse Gaussian distribution, and adding one more weight parameter . It is a special case for , where and is the weighted parameter. We investigate the properties of Generalized Crack distribution including first four moments, parameters estimation by using the maximum likelihood estimators and method of moment estimation. Evaluate the performance of the estimators by using bias. The results of simulation are presented in numerically and graphically.422021422021106111https://www.wseas.org/multimedia/journals/mathematics/2021/a225106-007(2021).pdf10.37394/23206.2021.20.11https://www.wseas.org/multimedia/journals/mathematics/2021/a225106-007(2021).pdfE. Schrodinger, “ Zur theorie der fall-und steigversuche an teilchenn mit brownscher bewegung, ” Physikalische Zeitschrift 16, 289–295 (1915).A. Wald, Sequential Analysis (New York, John Wiley and Sons, 1947)10.1214/aoms/1177706881M. C. K. Tweedie, “ Statistical properties of Inverse Gaussian distributions. I, II,” Ann. Math. Statist. 28 362–377, 696–705 (1957).J. Shuster, “ On the Inverse Gaussian distribution function. ” J. Amer. Statist. Assoc. 63, 1514–1516 (1968).R. S. Chhikara and J. L. Folks, The Inverse Gaussian Distribution: Theory, Methodology, and Applications (Marcel Dekker Inc., New York, 1989).10.1016/j.spl.2014.03.023Y. P. Chaubey, D. Sen, and K. K. Saha, “ On testing the coefficient of variation in an inverse Gaussian population. ” Statist. Probab. Lett. 90, 121–128 (2014).M. Ahsanullah and S. N. U. A. Kirmani, “ A characterization of the Wald distribution,” Naval Res. Logist. Quart. 31 (1), 155–158 (1984).10.1109/24.46490R. Khattree, “ Characterization of Inverse-Gaussian and gamma distributions through their length-biased distribution,” IEEE Reliabilitty 38, 610–611 (1989).G. P. Patil and C. R. Rao, Weighted distributions and a survey of their applications. In: Applications of Statistics, P. R. Krishnaiah, Eds., North-Holland Publishing Co., 383–405 (1977).B. Jørgensen, V. Seshadri, and G. A. Whitmore, “ On the mixture of the Inverse Gaussian distribution with its complementary reciprocal,” Scand. J. Statist. 18 (1), 77–89 (1991).10.1016/0378-3758(94)00148-oR. C. Gupta and H. O. Akman, “ On the reliability studies of a weighted Inverse Gaussian model,” J. Statist. Plann. Inference 48 (1), 69–83 (1995).R. C. Gupta and H. O. Akman, “ Bayes estimation in a mixture Inverse Gaussian model,” Ann. Inst. Statist. Math. 47 (3), 493–503 (1995).10.1007/bf02674095I. N. Volodin and O. A. Dzhungurova, On limit distributions emerging in the generalized Birnbaum–Saunders model. In: Proceedings of the 19th Seminar on Stability Problems for Stochastic Models, Part I (Vologda, 1998) 99, 1348–1366 (2000).M. Duangsaphon, Improved statistical inference for three-parameter Crack lifetime distribution. Ph. D. Thesis, Thammasat University (2014).10.3923/jas.2014.758.766P. Saengthong and W. Bodhisuwan, “ A new two-parameter Crack distribution,” Applied Sciences 14 (8), 758–766 (2014).10.1134/s1995080219080201Ngamkham. "On the Crack Random Numbers Generation Procedure," Lobachevskii Journal of Mathematics, (2019).10.46300/9106.2020.14.36Pulu Han, Nonlinear Mechanics Study of Concrete T-beam Bridge With Cracking Damage Based on Numerical Simulation, International Journal of Circuits, Systems and Signal Processing, Volume 14, 249-254 (2020).Z. Kala, A. Omishore, S. Seitl, M. Krejsa, J. Kala, Identification of Variation Coefficient of Equivalent Stress Range of Steel Girders with Cracks, International Journal of Mechanics, Volume 13, 69-78 (2019).