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

Part of: Survival analysis and censored data Operations research and management science

Published online by Cambridge University Press:  16 January 2019

Muhyiddin Izadi ,
Maryam Sharafi  and
Baha-Eldin Khaledi
Muhyiddin Izadi*
Affiliation:
Razi University
Maryam Sharafi*
Affiliation:
Razi University
Baha-Eldin Khaledi*
Affiliation:
Razi University Portland Sate University
*
* Postal address: Department of Statistics, Razi University, Kermanshah, Iran.
* Postal address: Department of Statistics, Razi University, Kermanshah, Iran.
*** Current address: Fariborz Maseeh Department of Mathematics and Statistics, Portland Sate University, Portland, OR 97201, USA.
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Abstract

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.

Keywords

BFR distributionconvex and concave functionsnonmonotonic ageing classNWBUE distributiontotal time on test transform

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1238 - 1248
Copyright
Copyright © Applied Probability Trust 2018 

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References

[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, 15–26.Google Scholar
[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. 451–481.Google Scholar
[3]Barlow, R. E. and Proschan, F. (1965). Mathematical Theory of Reliability. Wiley, New York.Google Scholar
[4]Barlow, R. E. and Proschan, F. (1966). Tolerance and confidence limits for classes of distributions based on failure rate. Ann. Math. Statist. 57, 1593–1601.Google Scholar
[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, 1–14.Google Scholar
[6]Bergamn, B. (1979). On age replacement and the total time on test concept. Scand. J. Statist. 4, 171–177.Google Scholar
[7]Bergman, B. and Klefsjö, B. (1984). The total time on test concept and its use in reliability theory. Operat. Res. 32, 596–606.Google Scholar
[8]Cha, J. H. and Mi, J. (2005). Optimal burn-in procedures in a generalized environment. Int. J. Reliab. Qual. Saf. Eng. 12, 189–201.Google Scholar
[9]Despande, J. V. and Suresh, R. P. (1990). Non-monotonic ageing. Scand. J. Statist. 17, 257–262.Google Scholar
[10]Embrechts, P. and Hofert, M. (2013). A note on generalized inverses. Math. Methods Operat. Res. 17, 423–432.Google Scholar
[11]Guess, F., Hollander, M. and Proschan, M. (1986). Testing exponentiality versus a trend change in mean residual life. Ann. Statist. 14, 1388–1398.Google Scholar
[12]Kattumannil, S. K. and Anisha, P. (2016). A simple non-parametric test for decreasing mean time to failure. Statist. Papers Available at https://doi.org/10.1007/s00362-016-0827-y.Google Scholar
[13]Kayid, M. and Izadkhah, S. (2014). Mean inactivity time function, associated orderings, and classes of life distributions. IEEE Trans. Reliab. 63, 593–602.Google Scholar
[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, 101–113.Google Scholar
[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, 670–678.Google Scholar
[16]Klefsjö, B. (1982). On aging properties and total time on test transforms. Scand. J. Statist. 9, 37–41.Google Scholar
[17]Klefsjö, B. (1989). Testing against a change in the NBUE property. Microelectron. Reliab. 29, 559–570.Google Scholar
[18]Knopik, L. (2005). Some results on ageing class. Control Cybern. 34, 1175–1180.Google Scholar
[19]Knopik, L. (2006). Characterization of a class of lifetime distributions. Control Cybern. 35, 407–414.Google Scholar
[20]Lai, C. D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.Google Scholar
[21]Lai, C. D., Xie, M. and Murthy, D. N. P. (2003). Modified Weibull model. IEEE Trans. Reliab. 52, 33–37.Google Scholar
[22]Li, X. and Xu, M. (2008). Reversed hazard rate order of equilibrium distributions and a related ageing notion. Statist. Papers 49, 749–767.Google Scholar
[23]Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.Google Scholar
[24]Mi, J. (1993). Discrete bathtub failure rate and upside-down bathtub mean residual life. Naval Res. Logistics 40, 361–371.Google Scholar
[25]Mi, J. (1995). Bathtub failure rate and upside-down bathtub mean residual life. IEEE Trans. Reliab. 44, 388–391.Google Scholar
[26]Mitra, M. and Basu, S. K. (1994). On a nonparametric family of life distributions and its dual. J. Statist. Plann. Inference 39, 385–397.Google Scholar
[27]Nakagawa, T. (2005). Maintenance Theory of Reliability. Springer, London.Google Scholar