Skip to main content Accessibility help

New nonparametric classes of distributions in terms of mean time to failure in age replacement

  • Muhyiddin Izadi (a1), Maryam Sharafi (a1) and Baha-Eldin Khaledi (a1) (a2)


The mean time to failure (MTTF) function in age replacement is used to evaluate the performance and effectiveness of the age replacement policy. In this paper, based on the MTTF function, we introduce two new nonparametric classes of lifetime distributions with nonmonotonic mean time to failure in age replacement; increasing then decreasing MTTF (IDMTTF) and decreasing then increasing MTTF (DIMTTF). The implications between these classes of distributions and some existing classes of nonmonotonic ageing classes are studied. The characterizations of IDMTTF and DIMTTF in terms of the scaled total time on test transform are also obtained.


Corresponding author

* Postal address: Department of Statistics, Razi University, Kermanshah, Iran.
** Email address:
*** Current address: Fariborz Maseeh Department of Mathematics and Statistics, Portland Sate University, Portland, OR 97201, USA.


Hide All
[1]Asha, G. and Nair, N. U. (2010). Reliability properties of mean time to failure in age replacement models. Int. J. Reliab. Qual. Saf. Eng. 17, 1526.
[2]Barlow, R. E. and Campo, R. (1975). Total time on test processes and applications to failure data analysis. In Reliability and Fault Tree Analysis, eds R. E. Barlow, J. Fussell and N. D. Singpurwalla, SIAM, Philadelphia, pp. 451481.
[3]Barlow, R. E. and Proschan, F. (1965). Mathematical Theory of Reliability. Wiley, New York.
[4]Barlow, R. E. and Proschan, F. (1966). Tolerance and confidence limits for classes of distributions based on failure rate. Ann. Math. Statist. 57, 15931601.
[5]Belzunce, F., Ortega, E. and Ruiz, J. M. (2007). On non-monotonic aging properties from the Laplace transform, with actuarial applications. Insurance Math. Econom. 40, 114.
[6]Bergamn, B. (1979). On age replacement and the total time on test concept. Scand. J. Statist. 4, 171177.
[7]Bergman, B. and Klefsjö, B. (1984). The total time on test concept and its use in reliability theory. Operat. Res. 32, 596606.
[8]Cha, J. H. and Mi, J. (2005). Optimal burn-in procedures in a generalized environment. Int. J. Reliab. Qual. Saf. Eng. 12, 189201.
[9]Despande, J. V. and Suresh, R. P. (1990). Non-monotonic ageing. Scand. J. Statist. 17, 257262.
[10]Embrechts, P. and Hofert, M. (2013). A note on generalized inverses. Math. Methods Operat. Res. 17, 423432.
[11]Guess, F., Hollander, M. and Proschan, M. (1986). Testing exponentiality versus a trend change in mean residual life. Ann. Statist. 14, 13881398.
[12]Kattumannil, S. K. and Anisha, P. (2016). A simple non-parametric test for decreasing mean time to failure. Statist. Papers Available at
[13]Kayid, M. and Izadkhah, S. (2014). Mean inactivity time function, associated orderings, and classes of life distributions. IEEE Trans. Reliab. 63, 593602.
[14]Kayid, M., Izadkhah, S. and Alshami, S. (2016). Laplace transform ordering of time to failure in age replacement models. J. Korean Stat. Soc. 45, 101113.
[15]Kayid, M., Ahmed, I. A., Izadkhah, S. and Abouammoh, A. M. (2013). Further results involving the mean time to failure order, and the decreasing mean time to failure class. IEEE Trans. Reliab. 62, 670678.
[16]Klefsjö, B. (1982). On aging properties and total time on test transforms. Scand. J. Statist. 9, 3741.
[17]Klefsjö, B. (1989). Testing against a change in the NBUE property. Microelectron. Reliab. 29, 559570.
[18]Knopik, L. (2005). Some results on ageing class. Control Cybern. 34, 11751180.
[19]Knopik, L. (2006). Characterization of a class of lifetime distributions. Control Cybern. 35, 407414.
[20]Lai, C. D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.
[21]Lai, C. D., Xie, M. and Murthy, D. N. P. (2003). Modified Weibull model. IEEE Trans. Reliab. 52, 3337.
[22]Li, X. and Xu, M. (2008). Reversed hazard rate order of equilibrium distributions and a related ageing notion. Statist. Papers 49, 749767.
[23]Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.
[24]Mi, J. (1993). Discrete bathtub failure rate and upside-down bathtub mean residual life. Naval Res. Logistics 40, 361371.
[25]Mi, J. (1995). Bathtub failure rate and upside-down bathtub mean residual life. IEEE Trans. Reliab. 44, 388391.
[26]Mitra, M. and Basu, S. K. (1994). On a nonparametric family of life distributions and its dual. J. Statist. Plann. Inference 39, 385397.
[27]Nakagawa, T. (2005). Maintenance Theory of Reliability. Springer, London.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed