Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-14T00:35:36.440Z Has data issue: false hasContentIssue false
  • English
  • Français

On classes of life distributions based on the mean time to failure function

Part of: Special processes Survival analysis and censored data

Published online by Cambridge University Press:  23 June 2021

Ruhul Ali Khan Open the ORCID record for Ruhul Ali Khan [Opens in a new window] ,
Dhrubasish Bhattacharyya  and
Murari Mitra
Ruhul Ali Khan*
Affiliation:
Indian Institute of Engineering Science and Technology, Shibpur
Dhrubasish Bhattacharyya*
Affiliation:
Indian Institute of Engineering Science and Technology, Shibpur
Murari Mitra*
Affiliation:
Indian Institute of Engineering Science and Technology, Shibpur
*
*Postal address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, PO Botanic Garden, Howrah 711103, West Bengal, India.
*Postal address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, PO Botanic Garden, Howrah 711103, West Bengal, India.
*Postal address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, PO Botanic Garden, Howrah 711103, West Bengal, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

The performance and effectiveness of an age replacement policy can be assessed by its mean time to failure (MTTF) function. We develop shock model theory in different scenarios for classes of life distributions based on the MTTF function where the probabilities $\bar{P}_k$ of surviving the first k shocks are assumed to have discrete DMTTF, IMTTF and IDMTTF properties. The cumulative damage model of A-Hameed and Proschan [1] is studied in this context and analogous results are established. Weak convergence and moment convergence issues within the IDMTTF class of life distributions are explored. The preservation of the IDMTTF property under some basic reliability operations is also investigated. Finally we show that the intersection of IDMRL and IDMTTF classes contains the BFR family and establish results outlining the positions of various non-monotonic ageing classes in the hierarchy.

Keywords

Age replacement policyIDMTTF classpure-birth processcumulative damage modelweak convergence

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 289 - 313
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A-Hameed, M. and Proschan, F. (1973). Nonstationary shock models. Stoch. Process. Appl. 1, 383–404.10.1016/0304-4149(73)90019-7CrossRefGoogle Scholar
A-Hameed, M. and Proschan, F. (1975). Shock models with underlying birth process. J. Appl. Prob. 12, 18–28.10.2307/3212403CrossRefGoogle Scholar
Abouammoh, A., Hindi, M. and Ahmed, A. (1988). Shock models with NBUFR and NBAFR survivals. Trabajos de Estadistica 3, 97–113.10.1007/BF02863509CrossRefGoogle Scholar
Anderson, K. K. (1987). Limit theorems for general shock models with infinite mean intershock times. J. Appl. Prob. 24, 449–456.10.2307/3214268CrossRefGoogle Scholar
Anis, M. (2012). On some properties of the IDMRL class of life distributions. J. Statist. Planning Infer. 142, 3047–3055.CrossRefGoogle Scholar
Asha, G. and Nair, U. (2010). Reliability properties of mean time to failure in age replacement models. Internat. J. Reliab. Quality Safety Eng. 17, 15–26.10.1142/S0218539310003640CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1965). Mathematical Theory of Reliability. Wiley, New York.Google Scholar
Barlow, R. E. and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD.Google Scholar
Basu, S. K. and Bhattacharjee, M. C. (1984). On weak convergence within the HNBUE family of life distributions. J. Appl. Prob. 21, 654–660.10.2307/3213627CrossRefGoogle Scholar
Basu, S. K. and Simons, G. (1982). Moment spaces for IFR distributions, applications and related material. In Contributions to Statistics: Essays in Honour of Norman L. Johnson, ed. P. K. Sen. North-Holland, Amsterdam.Google Scholar
Belzunce, F., Ortega, E.-M. and Ruiz, J. M. (2007). On non-monotonic ageing properties from the Laplace transform, with actuarial applications. Insurance Math. Econom. 40, 1–14.10.1016/j.insmatheco.2006.01.010CrossRefGoogle Scholar
Berger, R. L., Boos, D. D. and Guess, F. M. (1988). Tests and confidence sets for comparing two mean residual life functions. Biometrics 44, 103–115.CrossRefGoogle ScholarPubMed
Bhattacharyya, D., Khan, R. A. and Mitra, M. (2020). A test of exponentiality against DMTTF alternatives via L-statistics. Statist. Prob. Lett. 165, 1–9.10.1016/j.spl.2020.108853CrossRefGoogle Scholar
Bhattacharyya, D., Khan, R. A. and Mitra, M. (2020). A nonparametric test for comparison of mean past lives. Statist. Prob. Lett. 161. Available at https://doi.org/10.1016/j.spl.2020.108722.CrossRefGoogle Scholar
Block, H. W. and Savits, T. H. (1978). Shock models with NBUE survival. J. Appl. Prob. 15, 621–628.10.2307/3213125CrossRefGoogle Scholar
Deshpande, J. V. and Suresh, R. P. (1990). Non-monotonic ageing. Scand. J. Statist. 17, 257–262.Google Scholar
Ebrahimi, N. (1999). Stochastic properties of a cumulative damage threshold crossing model. J. Appl. Prob. 36, 720–732.10.1239/jap/1032374629CrossRefGoogle Scholar
Esary, J., Marshall, A. and Proschan, F. (1973). Shock models and wear processes. Ann. Prob. 1, 627–649.10.1214/aop/1176996891CrossRefGoogle Scholar
Fagiuoli, E. and Pellerey, F. (1994). Preservation of certain classes of life distributions under Poisson shock models. J. Appl. Prob. 31, 458–465.10.2307/3215038CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. I. John Wiley, New York.Google Scholar
Finkelstein, M. and Cha, J. H. (2013). Stochastic modeling for reliability. In Shocks, Burn-in and Heterogeneous Populations (Springer Series in Reliability Engineering). Springer, London.Google Scholar
Ghosh, M. and Ebrahimi, N. (1982). Shock models leading to increasing failure rate and decreasing mean residual life survival. J. Appl. Prob. 19, 158–166.10.2307/3213925CrossRefGoogle Scholar
Glaser, R. E. (1980). Bathtub and related failure rate characterizations. J. Amer. Statist. Assoc. 75, 667–672.10.1080/01621459.1980.10477530CrossRefGoogle Scholar
Guess, F., Hollander, M. and Proschan, F. (1986). Testing exponentiality versus a trend change in mean residual life. Ann. Statist. 14, 1388–1398.10.1214/aos/1176350165CrossRefGoogle Scholar
Izadi, M. and Fathimanesh, S. (2019). Testing exponentiality against a trend change in mean time to failure in age replacement. Commun. Statist. Theory Methods. Available at https://doi.org/10.1080/03610926.2019.1702693.CrossRefGoogle Scholar
Izadi, M., Sharafi, M. and Khaledi, B.-E. (2018). New nonparametric classes of distributions in terms of mean time to failure in age replacement. J. Appl. Prob. 55, 1238–1248.CrossRefGoogle Scholar
Kattumannil, S. K. and Anisha, P. (2019). A simple non-parametric test for decreasing mean time to failure. Statist. Papers 60, 73–87.CrossRefGoogle Scholar
Kayid, M., Ahmad, I., Izadkhah, S. and Abouammoh, A. (2013). Further results involving the mean time to failure order, and the decreasing mean time to failure class. IEEE Trans. Reliab. 62, 670–678.10.1109/TR.2013.2270423CrossRefGoogle Scholar
Khan, R. A. and Mitra, M. (2019). Sharp bounds for survival probability when ageing is not monotone. Prob. Eng. Inf. Sci. 33, 205–219.10.1017/S0269964818000128CrossRefGoogle Scholar
Khan, R. A., Bhattacharyya, D. and Mitra, M. (2020). A change point estimation problem related to age replacement policies. Operat. Res. Lett. 48, 105–108.10.1016/j.orl.2019.12.005CrossRefGoogle Scholar
Klefsjö, B. (1981). HNBUE survival under some shock models. Scand. J. Statist. 8, 39–47.Google Scholar
Klefsjö, B. (1982). On aging properties and total time on test transforms. Scand. J. Statist. 9, 37–41.Google Scholar
Klefsjö, B. (1983). A useful ageing property based on the Laplace transform. J. Appl. Prob. 20, 615–626.10.2307/3213897CrossRefGoogle Scholar
Klefsjö, B. (1989). Testing against a change in the NBUE property. Microelectron. Reliab. 29, 559–570.10.1016/0026-2714(89)90346-6CrossRefGoogle Scholar
Knopik, L. (2005). Some results on the ageing class. Control Cybernet. 34, 1175–1180.Google Scholar
Knopik, L. (2006). Characterization of a class of lifetime distributions. Control Cybernet. 35, 407–414.Google Scholar
Kochar, S. C. (1990). On preservation of some partial orderings under shock models. Adv. Appl. Prob. 22, 508–509.10.2307/1427555CrossRefGoogle Scholar
Lai, C. D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York.Google Scholar
Li, X. and Xu, M. (2008). Reversed hazard rate order of equilibrium distributions and a related aging notion. Statist. Papers 49, 749–767.CrossRefGoogle Scholar
Loève, M. (1963). Probability Theory, 3rd edn. Van Nostrand, New York.Google Scholar
Mitra, M. and Basu, S. K. (1994). On a nonparametric family of life distributions and its dual. J. Statist. Planning Infer. 39, 385–397.10.1016/0378-3758(94)90094-9CrossRefGoogle Scholar
Mitra, M. and Basu, S. K. (1996). On some properties of the bathtub failure rate family of life distributions. Microelectron. Reliab. 36, 679–684.10.1016/0026-2714(95)00136-0CrossRefGoogle Scholar
Mitra, M. and Basu, S. K. (1996). Shock models leading to non-monotonic ageing classes of life distributions. J. Statist. Planning Infer. 55, 131–138.10.1016/0378-3758(95)00192-1CrossRefGoogle Scholar
Nakagawa, T. (2007). Shock and Damage Models in Reliability Theory. Springer, London.Google Scholar
Park, D. H. (2003). Class of NBU-$t_0$ life distribution. In Handbook of Reliability Engineering, ed. H. Pham, pp. 181–197. Springer, London.Google Scholar
Pellerey, F. (1993). Partial orderings under cumulative damage shock models. Adv. Appl. Prob. 25, 939–946.10.2307/1427800CrossRefGoogle Scholar
Pérez-Ocón, R. and Gámiz-Pérez, M. L. (1995). On the HNBUE property in a class of correlated cumulative shock models. Adv. Appl. Prob. 27, 1186–1188.10.2307/1427938CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, NewYork.10.1007/978-0-387-34675-5CrossRefGoogle Scholar
Shanthikumar, J. G. and Sumita, U. (1983). General shock models associated with correlated renewal sequences. J. Appl. Prob. 20, 600–614.10.2307/3213896CrossRefGoogle Scholar
Sudheesh, K. K., Asha, G. and Krishna, K. M. J. (2019). On the mean time to failure of an age-replacement model in discrete time. Commun. Statist. Theory Meth. Available at https://doi.org/10.1080/03610926.2019.1672742.Google Scholar
Suresh, R. (2003). On the inter-relationships between some classes of life distributions. Calcutta Statist. Assoc. Bull. 54, 261–268.10.1177/0008068320030310CrossRefGoogle Scholar
Yamada, K. (1989). Limit theorems for jump shock models. J. Appl. Prob. 26, 793–806.10.2307/3214384CrossRefGoogle Scholar