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Testing of UBAC(2) Class of Life Distributions based on TTT - Transform

International Journal of Computer Applications
Foundation of Computer Science (FCS), NY, USA
Year of Publication: 2020
S.E. Abu-Youssef, A.A. El-Toony

S E Abu-Youssef and A A El-Toony. Testing of UBAC(2) Class of Life Distributions based on TTT - Transform. International Journal of Computer Applications 175(29):9-12, November 2020. BibTeX

	author = {S.E. Abu-Youssef and A.A. El-Toony},
	title = {Testing of UBAC(2) Class of Life Distributions based on TTT - Transform},
	journal = {International Journal of Computer Applications},
	issue_date = {November 2020},
	volume = {175},
	number = {29},
	month = {Nov},
	year = {2020},
	issn = {0975-8887},
	pages = {9-12},
	numpages = {4},
	url = {},
	doi = {10.5120/ijca2020920820},
	publisher = {Foundation of Computer Science (FCS), NY, USA},
	address = {New York, USA}


In this paper, a new test statistic for testing exponentiality against used better than aged in increasing concave ordering UBAC(2) is constructed based on total time on test (TTT)-Transform. Critical values are tabulated for sample size n = 10(5)100. The power of the test is estimated for some commonly used distributions in reliability. Finally, medical applications for real data are proposed to illustrate the theoretical results.


  1. Bryson, M. C. and Siddiqui, M. M. 1969. Some criteria for aging. Journal of the American Statistical Association, Vol. 64, No. 328, pp. 1472-1483.
  2. Barlow, R. E. and Proschan, F. 1981. Mathematical theory of reliability. To Begin With: Silver-Spring, MD.
  3. Abu-Youssef, S. E. 2004. Non-parametric Test for Monotone Variance Residual Life Class of Life Distributions with Hypothesis Testing Applications. Applied Mathematics and Computations, Vol. 158, No. 3, pp. 817-826.
  4. Ahmed, I. A. 2004. Some Properties of Classes of Life Distributions with Unknown Age. Statistics and Probability Letters, Vol. 69, No. 3, pp. 333-342.
  5. Barlow, R. E. and Campo, R. 1975. Total Time on Test Processes and Application to Failure Data Analysis. Reliab. Fault Tree Analysis, SIAM, Philadephia, Vol. 8, No. 4, pp. 451-481.
  6. Klefsjo, B. 1980. Some Aging Properties and The Total Time on Test Transform. Res. Rep., Dept. of Math. Statist. Univ. of Umea, Sweden.
  7. Klefsjo, B. 1982. The HNBUE and HNWUE classes of life distributions. Naval Logist Res. Quart, Vol. 29, pp. 331-344.
  8. Klefsjo, B. 1983. Some Tests Against Aging Based on Total Time on Test Transform. Comm. Statist., Theor. Meth., Vol.12, No. 8, pp. 907-927.
  9. Bergman, B. and Klefsjo, B. 1984. The total time on test concept and its use in reliability theory. Operat. Res., Vol. 32, No. 3, pp. 596-606.
  10. Pham, T. G. and Turkkan, M. 1994. The Lorenz and the scaled total-time-on-test transform curves: a unified approach. IEEE Trans. Reliab., Vol. 43, No. 1, PP. 76-84.
  11. Bartoszewicz, J. 1995. Stochastic order relations and the total time on test transform.
  12. S
  13. tatist. Prob. Lett., Vol. 22, No. 2, pp. 103-110.
  14. Bartoszewicz, J. 1996. Tail orderings and the total time on test transform. Appl. Math., Vol, 24, No. 1,pp. 77-86.
  15. Haupt, E. and Schabe, H. 1997. The TTT transformation and a new bathtab distribution model. J. Statist. Planning Infer., Vol. 60, No. 2, pp. 229-240.
  16. Kochar, S. C., Li, X., and Shaked, M. 2002. The total time on test transform and the excess wealth stochastic orders of distributions. Adv. Appl. Prob., Vol. 34, No. 4, pp. 826-845.
  17. Li, X. and Zou, M. 2004. Preservation of stochastic orders for random minima and maxima with applications. Naval Res. Logistics., Vol. 51, No. 3, pp. 332-334.
  18. Ahmed, I. A., Li, X. and Kayid, M. 2005. The NBUT class of life distributions. IEEE Trans. Reliab., Vol. 54, No. 3, pp. 396-401.
  19. Li, H. and Shaked, M. 2007. A general family of univariate stochastic orders. J. Statist. Planning Infer., Vol. 137, No. 11, pp. 3601-3610.
  20. Al-Nachawati, H. 2007. Test for Monotone Variance Residual Life Class of Life Distributions Based on Total Time on Test Transformation. J. King Saud Univ., Vol. 19, No. 2, pp. 109-117.
  21. Nanda, A. K. and Shaked, M. 2008. Partial ordering and aging properties of order statistics when sample size is random: a brief review. Commun. Statist. Theory Meth., Vol. 37, No. 11, pp. 1710-1720.
  22. Abu-Youssef, S. E., Mohie El-Din, M. M. and Hassan, M. KH. 2012. Testing Of EBELC Classes Of Life Distributions Based On TTT - Transform. International Journal of Reliability and Applications., Vol. 13, No. 1, pp. 49-56.
  23. Mohie El-Din, M. M., Abu-Youssef, S. E. and 2013. Testing unknown age classes of life distributions based on TTT-transform. International Journal of Reliability and Applications., Vol. 14, No. 1, pp. 1-9.
  24. Ali, N. S. A. 2018. On the Properties of the UBAC(2) Class of Life Distributions. Journal of Testing and Evaluation., Vol. 46, No. 2, pp.730-735.
  25. Willmot, G. and Cai, J. 2000. On classes of life time distributions with unknown age. Probability in the engineering and informational sciences, Vol. 14, No. 4, pp. 473-484.
  26. Dicrescenzo, A. 1999. Dual stochastic ordering, decreasing aging properties of divices of unknown age. it Communications in Statistics. Stochastic Models, Vol. 15, No. 3, pp. 561-576.
  27. Mohie El-Din, M. M., Abu-Youssef, S. E. and Ali, N. S. A. 2015. A New Class of Life Distributions Based On Unknown Age, IJRA, Vol. 16, No. 1, pp. 27–34.
  28. Deshpand, J. V., Kocher, S. C. and Singh, H. 1986. Aspects of Positive Aging. J. Appl. Probab., Vol. 28, No. 3, pp. 1472-1483.
  29. Deshpand, J. V. and Purohit, S. G. 2005. Life Time: Statistical Models and Methods, World Scientific Publishing Co., Singapore. Vol. 11.
  30. Barlow, R. E. and Doksum, K. 1972. Isotonic tests for convex ordering. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Prob. Univ. of Calif., Vol. 1, pp. 293-323.
  31. Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. 1972. Statistical inference under order restriction. Wiley, New York, Vol. 27, No. 4, pp. 139-189.
  32. Barlow, R. E. 1979. Geometry of the total time on test transform, Naval Res. Logist. Quart., Vol. 26, No. 3, pp. 393-402.
  33. Hollander, M. and Prochan, F. 1975. Test for mean residual life. Biometrika. 62(3), pp. 585-593.
  34. Abu-Youssef, S. E. 2009. A Goodness of Fit Approach to Monotone Variance Residual LifeClass of Life Distributions, Applied Mathematical Sciences., Vol. 3, no. 15, 715-724.
  35. Kochar, S. C. 1985. Testing exponentiality against monotone failure rate average. Comm. in Stat. - Theo. and Meth., Vol. 14, No. 2, pp. 381-392.
  36. Attia, A. F., Mahmoud, M. A. W. and Abdul-Moniem, I. B. 2004. On Testing for Exponential Better than Used in Average Class of Life Distributions Based on the U-Test. The proceeding of The 39 th Annual Conference on Statistics. Computer Sciences and Operation Research, ISSR Cairo University-Egypt, pp. 11-14.


UBAC(2) classes of life distributions; Survival function; Exponentiality; Total time on test (TTT)-transform; Monte Carlo method.