# Total Time on Test Transforms

• N. Unnikrishnan Nair
• P. G. Sankaran
• N. Balakrishnan
Chapter
Part of the Statistics for Industry and Technology book series (SIT)

## Abstract

The total time on test transform is essentially a quantile-based concept developed in the early 1970s. Apart from its applications in reliability problems, it has also been found useful in other areas like stochastic modelling, maintenance scheduling, risk assessment of strategies and energy sales. When several units are tested for studying their life lengths, some of the units would fail while others may survive the test period. The sum of all observed and incomplete life lengths is the total time on test statistic. As the number of units on test tends to infinity, the limit of this statistic is called the total time on test transform (TTT). The definitions and properties of these two concepts are discussed and the functional forms of TTT for several life distributions are presented in Table 5.1. We discuss the Lorenz curve, Bonferroni curve and the Leimkuhler curve which are closely related to the TTT. Identities connecting various curves, characterizations of distributions in terms of these curves and their relationships with various reliability functions are detailed subsequently. In view of the ageing classes in the quantile set-up introduced in Chap. 4, it is possible to characterize these classes in terms of TTT. Accordingly, we give necessary and sufficient conditions for IHR, IHRA, DMRL, NBU, NBUE, HNBUE, NBUHR, NBUHRA, IFHA*t 0, UBAE, DMRLHA, DVRL, and NBU-t 0 classes in terms of the total time on test transform. Another interesting property of the TTT is that it uniquely determines the lifetime distribution. There have been several generalizations of the TTT. We discuss these extensions and their properties, with special reference to the TTT of order n. Relationships between the reliability functions of the baseline model and those of the TTT of order n (which is also a quantile function) are described and then utilized to describe the pattern of ageing of the transformed distributions. Some life distributions are characterized. The discussion of the applications of TTT in modelling includes derivation of the L-moments and other descriptive measures of the original distribution. Some of the areas in reliability engineering that widely use TTT are preventive maintenance, availability, replacement problems and burn-in strategies.

## Keywords

Preventive Maintenance Gini Index Lorenz Curve Quantile Function Empirical Distribution Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Aarset, M.V.: The null distribution of a test of constant versus bathtub failure rate. Scand. J. Stat. 12, 55–62 (1985)
2. 9.
Abouammoh, A.M., Khalique, A.: Some tests for mean residual life criteria based on the total time on test transform. Reliab. Eng. 19, 85–101 (1997)
3. 25.
Ahmad, I.A., Li, X., Kayid, M.: The NBUT class of life distributions. IEEE Trans. Reliab. 54, 396–401 (2005)
4. 39.
Asha, G., Nair, N.U.: Reliability properties of mean time to failure in age replacement models. Int. J. Reliab. Qual. Saf. Eng. 17, 15–26 (2010)
5. 60.
Balakrishnan, N., Sarabia, J.M., Kolev, N.: A simple relation between the Leimkuhler curve and the mean residual life function. J. Informetrics 4, 602–607 (2010)
6. 64.
Barlow, R.E.: Geometry of the total time on test transforms. Nav. Res. Logist. Q. 26, 393–402 (1979)
7. 65.
Barlow, R.E., Bartholomew, D.J., Bremner, J.M., Brunk, H.D.: Statistical Inference Under Order Restrictions. Wiley, New York (1972)
8. 66.
Barlow, R.E., Campo, R.: Total time on test process and applications to fault tree analysis. In: Reliability and Fault Tree Analysis, pp. 451–481. SIAM, Philadelphia (1975)Google Scholar
9. 67.
Barlow, R.E., Doksum, K.A.: Isotonic tests for convex orderings. In: The Sixth Berkeley Symposium in Mathematical Statistics and Probability I, Statistical Laboratory of the University of California, Berkeley, pp. 293–323, 1972Google Scholar
10. 73.
Bartoszewicz, J.: Stochastic order relations and the total time on test transforms. Stat. Probab. Lett. 22, 103–110 (1995)
11. 74.
Bartoszewicz, J.: Tail orderings and the total time on test transforms. Applicationes Mathematicae 24, 77–86 (1996)
12. 75.
Bartoszewicz, J.: Application of a general composition theorem to the star order of distributions. Stat. Probab. Lett. 38, 1–9 (1998)
13. 76.
Bartoszewicz, J., Benduch, M.: Some properties of the generalized TTT transform. J. Stat. Plann. Infer. 139, 2208–2217 (2009)
14. 89.
Bergman, B.: Crossings in the total time on test plot. Scand. J. Stat. 4, 171–177 (1977)
15. 90.
Bergman, B.: On age replacement and total time on test concept. Scand. J. Stat. 6, 161–168 (1979)
16. 91.
Bergman, B.: On the decision to replace a unit early or late: A graphical solution. Microelectron. Reliab. 20, 895–896 (1980)
17. 92.
Bergman, B., Klefsjö, B.: The TTT-transforms and age replacements with discounted costs. Nav. Res. Logist. Q. 30, 631–639 (1983)
18. 93.
Bergman, B., Klefsjö, B.: The total time on test and its uses in reliability theory. Oper. Res. 32, 596–606 (1984)
19. 94.
Bergman, B., Klefsjö, B.: Burn-in models and TTT-transforms. Qual. Reliab. Eng. Int. 1, 125–130 (1985)
20. 95.
Bergman, B., Klefsjö, B.: The TTT-concept and replacements to extend system life. Eur. J. Oper. Res. 28, 302–307 (1987)
21. 96.
Bergman, B., Klefsjö, B.: A family of test statistics for detecting monotone mean residual life. J. Stat. Plann. Infer. 21, 161–178 (1989)
22. 125.
Campo, R.A.: Probabilistic optimality in long-term energy sales. IEEE Trans. Power Syst. 17, 237–242 (2002)
23. 132.
Chan, P.K.W., Downs, T.: Two criteria for preventive maintenance. IEEE Trans. Reliab. 27, 272–273 (1968)Google Scholar
24. 133.
Chandra, M., Singpurwalla, N.D.: The Gini index, Lorenz curve and the total time on test transform. Technical Report, George Washington University, Washington, DC, 1978Google Scholar
25. 134.
Chandra, M., Singpurwalla, N.D.: Relationships between some notions which are common to reliability theory and economics. Math. Oper. Res. 6, 113–121 (1981)
26. 151.
Cleroux, P., Dubuc, S., Tilquini, C.: The age replacement problem with minimal repair costs. Oper. Res. 27, 1158–1167 (1979)
27. 161.
Csorgo, M., Yu, H.: Estimation of total time on test transforms for stationary observations. Stoch. Proc. Appl. 68, 229–253 (1997)
28. 171.
Derman, C., Lieberman, G.J., Ross, S.M.: On the use of replacements to extend system life. Oper. Res. 32, 616–627 (1984)
29. 180.
Dohi, T., Kiao, N., Osaki, S.: Solving problems of a repairable limit using TTT concept. IMA J. Math. Appl. Bus. Ind. 6, 101–115 (1995)
30. 183.
Ebrahimi, N., Spizzichino, F.: Some results on normalised total time on test and spacings. Stat. Probab. Lett. 36, 231–243 (1997)
31. 218.
Giorgi, G.M.: Concentration index, Bonferroni. In: Encyclopedia of Statistical Sciences, vol. 2, pp. 141–146. Wiley, New York (1998)Google Scholar
32. 219.
Giorgi, G.M., Crescenzi, M.: A look at the Bonferroni inequality measure in a reliability framework. Statistica 61(4), 571–583 (2001)
33. 266.
Haupt, E., Schäbe, H.: The TTT transformation and a new bathtub distribution model. J. Stat. Plann. Infer. 60, 229–240 (1997)
34. 318.
Kayid, M.: A general family of NBU class of life distributions. Stat. Meth. 4, 1895–1905 (2007)
35. 333.
Klefsjö, B.: HNBUE and HNWUE classes of life distributions. Nav. Res. Logist. Q. 29, 615–626 (1982)
36. 334.
Klefsjö, B.: On ageing properties and total time on test transforms. Scand. J. Stat. 9, 37–41 (1982)
37. 335.
Klefsjö, B.: Some tests against ageing based on the total time on test transform. Comm. Stat. Theor. Meth. 12, 907–927 (1983)
38. 336.
Klefsjö, B.: Testing exponentiality against HNBUE. Scand. J. Stat. 10, 67–75 (1983)Google Scholar
39. 338.
Klefsjö, B.: TTT-transforms: A useful tool when analysing different reliability problems. Reliab. Eng. 15, 231–241 (1986)
40. 339.
Klefsjö, B.: TTT-plotting: A tool for both theoretical and practical problems. J. Stat. Plann. Infer. 29, 99–110 (1991)
41. 340.
Klefsjö, B., Westberg, U.: TTT plotting and maintenance policies. Qual. Eng. 9, 229–235 (1996–1997)Google Scholar
42. 341.
Kleiber, C., Kotz, S.: Statistical Size Distributions in Economics and Actuarial Sciences. Wiley, Hoboken (2003)
43. 348.
Kochar, S.C., Deshpande, J.V.: On exponential scores for testing against positive ageing. Stat. Probab. Lett. 3, 71–73 (1985)
44. 357.
Kumar, D., Westberg, U.: Maintenance scheduling under age replacement policy using proportional hazards model and TTT-plotting. Eur. J. Oper. Res. 99, 507–515 (1997)
45. 365.
Kvaloy, J.T., Lindqvist, B.H.: TTT based tests for trend in repairable systems data. Reliab. Eng. Syst. Saf. 60, 13–28 (1998)
46. 388.
Li, H., Shaked, M.: A general family of univariate stochastic orders. J. Stat. Plann. Infer. 137, 3601–3610 (2007)
47. 447.
Nair, N.U., Sankaran, P.G., Vineshkumar, B.: Total time on test transforms and their implications in reliability analysis. J. Appl. Probab. 45, 1126–1139 (2008)
48. 468.
Neath, A.A., Samaniego, F.J.: On the total time on test transform of an IFRA distribution. Stat. Probab. Lett. 14, 289–291 (1992)
49. 492.
Perez-Ocon, R., Gamiz-Perez, M.L., Ruiz-Castro, J.E.: A study of different ageing classes via total time on test transform and Lorenz curves. Appl. Stoch. Model. Data Anal. 13, 241–248 (1998)
50. 493.
Pham, T.G., Turkkan, M.: The Lorenz and the scaled total-time-on-test transform curves, A unified approach. IEEE Trans. Reliab. 43, 76–84 (1994)
51. 498.
Pundir, S., Arora, S., Jain, K.: Bonferroni curve and the related statistical inference. Stat. Probab. Lett. 75, 140–150 (2005)
52. 518.
Sarabia, J.M.: A general definition of Leimkuhler curves. J. Informetrics 2, 156–163 (2008)
53. 519.
Sarabia, J.M., Prieto, F., Sarabia, M.: Revisiting a functional form for the Lorenz curve. Econ. Lett. 105, 61–63 (2010)
54. 573.
Vera, F., Lynch, J.: K-Mart stochastic modelling using iterated total time on test transforms. In: Modern Statistical and Mathematical Methods in Reliability, pp. 395–409. World Scientific, Singapore (2005)Google Scholar
55. 578.
Westberg, U., Klefsjö, B.: TTT plotting for censored data based on piece-wise exponential estimator. Int. J. Reliab. Qual. Saf. Eng. 1, 1–13 (1994)
56. 579.
Wie, X.: Test of exponentiality against a monotone hazards function alternative based on TTT transformations. Microelectron. Reliab. 32, 607–610 (1992)
57. 580.
Wie, X.: Testing whether one distribution is more IFR than another. Microelectron. Reliab. 32, 271–273 (1992)
58. 592.
Xie, M.: Testing constant failure rate against some partially monotone alternatives. Microelectron. Reliab. 27, 557–565 (1987)
59. 593.
Xie, M.: Some total time on test quantities useful for testing constant against bathtub shaped failure rate distributions. Scand. J. Stat. 16, 137–144 (1989)
60. 601.
Zhao, N., Song, Y.H., Lu, H.: Risk assessment strategies using total time on test transforms. IEEE (2006). doi: 10.1109/PES.2006.1709062Google Scholar