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SOME SUFFICIENT CONDITIONS FOR RELATIVE AGING OF LIFE DISTRIBUTIONS

Published online by Cambridge University Press:  13 September 2016

Neeraj Misra ,
Jisha Francis  and
Sameen Naqvi
Neeraj Misra
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208016, India E-mail: neeraj@iitk.ac.in; jisha@iitk.ac.in; nsameen@iitk.ac.in
Jisha Francis
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208016, India E-mail: neeraj@iitk.ac.in; jisha@iitk.ac.in; nsameen@iitk.ac.in
Sameen Naqvi
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur-208016, India E-mail: neeraj@iitk.ac.in; jisha@iitk.ac.in; nsameen@iitk.ac.in
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Abstract

In reliability theory, Cox's proportional hazard model is quite popular and widely used. In many situations, it is observed that failure rates under consideration are not proportional, rather they cross each other. In such situations, an alternative to Cox's proportional hazard model may be monotone hazard ratio model (provided the ratio exists). A notion of relative aging based on increasing hazard ratio was introduced by Kalashnikov and Rachev [19]. Sengupta and Deshpande [40] further explored this model and posited two other notions of relative aging based on increasing reversed failure rate ratio and increasing mean residual life ratio. In this study, for two life distributions, we derive sufficient conditions under which a life distribution ages faster than the other with respect to notions of relative aging described above. These sufficient conditions are easy to verify and can be used in practical applications where one is interested in studying relative aging of two life distributions. Applications of these results to relative aging of weighted distributions have also been illustrated. We also introduce a new relative aging ordering in terms of mean inactivity time order and study its fundamental properties.

Keywords

applied probabilityreliability theory
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 31 , Issue 1 , January 2017 , pp. 83 - 99
Copyright
Copyright © Cambridge University Press 2016 

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References

1. Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing: probability models. New York: Holt, Rinehart and Winston.Google Scholar
2. Bartoszewicz, J. & Skolimowska, M. (2006). Preservation of classes of life distribution and stochastic orders under weighting. Statistics and Probability Letters 76: 587–596.CrossRefGoogle Scholar
3. Begg, C.B., McGlave, P.B., Bennet, J.M., Cassileth, P.A., & Oken, M.M. (1984). A critical comparison of allogeneic bone marrow transplantation and conventional chemotherapy as treatment for acute non-lymphomytic leukemia. Journal of Clinical Oncology 2: 369–378.CrossRefGoogle ScholarPubMed
4. Belzunce, F., Candel, J., & Ruiz, J.M. (1998). Ordering and asymptotic properties of residual income distributions. Sankhya 60: 331–348.Google Scholar
5. Boland, P.J., Proschan, F., & Tong, Y.L. (1992). A stochastic ordering of partial sums of independent random variables and of some random processes. Journal of Applied Probability 29: 645–654.Google Scholar
6. Boland, P.J., El-Neweihi, E., & Proschan, F. (1994). The hazard rate ordering with applications in reliability and order statistics. Journal of Applied Probability 31: 180–192.CrossRefGoogle Scholar
7. Blumenthal, S. (1963). Proportional sampling in life length studies. Technometrics 9: 205–218.CrossRefGoogle Scholar
8. Champlin, R., Mitsuyasu, R., Elashoff, R., & Gale, R.P. (1983). Recent advances in bone-marrow transplantation. In Gale, R.P. (ed.), UCLA Symposia on Molecular and Cellular Biology, Vol. 7, New York: Alan R. Liss, pp. 141–158.Google Scholar
9. Crescenzo, A.D. (2000). Some results on the proportional reversed hazards model. Statistics and Probability Letters 50: 313–321.Google Scholar
10. Deshpande, J.V., Kochar, S.C., & Singh, H. (1986). Aspects of positive ageing. Journal of Applied Probability 23: 748–758.Google Scholar
11. Eaton, M.L. (1982). A review of selected topics in multivariate probability inequalities. Annals of Statistics 10: 11–43.Google Scholar
12. Finkelstein, M. (2006). On relative ordering of mean residual lifetime functions. Statistics and Probability Letters 76: 939–944.Google Scholar
13. Fisher, R.A. (1934). The effects of methods of ascertainment upon the estimation of frequencies. Annals of Eugenics 6: 13–25.CrossRefGoogle Scholar
14. Block, H.W., Savits, T.H., & Singh, H. (1998). The reversed hazard rate function. Probability in the Engineering and Informational Sciences 12: 69–90.CrossRefGoogle Scholar
15. Jain, K., Singh, H., & Bagai, I. (1989). Relations for reliability measures of weighted distributions. Communications in Statistics—Theory and Methods 18: 4393–4412.Google Scholar
16. Gupta, R.C. & Keating, J.P. (1986). Relations for reliability measures under length biased sampling. Scandinavian Journal of Statistics 13: 49–56.Google Scholar
17. Gupta, R.C. & Kirmani, S.N.U.A. (1990). The role of weighted distributions in stochastic modelling. Communications in Statistics—Theory and Methods 19: 3147–3162.Google Scholar
18. Gupta, N., Misra, N., & Kumar, S. (2015). Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components. European Journal of Operational Research 240: 425–430.CrossRefGoogle Scholar
19. Kalashnikov, V.V. & Rachev, S.T. (1986). A characterization of queueing models and its stability. In Prohorov, Yu. K. et al. (eds.), Probability theory and mathematical statistics, pp. 37–53.Google Scholar
20. Kayid, M., Izadkhah, S., & Alshami, S. (2014). Residual probability function, associated orderings, and related aging classes. Mathematical Problems in Engineering 2014: Article ID 490692, 10 pp.Google Scholar
21. Kayid, M., Izadkhah, S., & Zuo, M.J. (2015). Some results on the relative ordering of two frailty models. Statistical Papers pp. 1–15.Google Scholar
22. Lai, C. & Xie, M. (2003). Relative ageing for two parallel systems and related problems. Mathematical and Computer Modelling 38: 1339–1345.CrossRefGoogle Scholar
23. Lai, C.D. & Xie, M. (2006). Stochastic ageing and dependence for reliability. New York, NY: Springer Science & Business Media.Google Scholar
24. Li, H. & Li, X. (2013). Stochastic orders in reliability and risk. In Honor of Professor Moshe Shaked. New York: Springer.Google Scholar
25. Li, C., & Li, X. (2016). Relative ageing of series and parallel systems with statistically independent and heterogeneous component lifetimes. IEEE Transactions on Reliability 99: 1–8.Google Scholar
26. Ng'andu, N.H. (1997). An empirical comparison of statistical tests for assessing the proportional hazards assumption of Cox's model. Statistics in Medicine 16: 611–626.3.0.CO;2-T>CrossRefGoogle ScholarPubMed
27. Nanda, A.K. & Jain, K. (1999). Some weighted distribution results on univariate and bivariate cases. Journal of Statistical Planning and Inference 77: 169–180.Google Scholar
28. Nanda, A.K., Das, S., & Balakrishnan, N. (2013). On dynamic proportional mean residual life model. Probability in the Engineering and Informational Sciences 27: 553–588.CrossRefGoogle Scholar
29. Marshall, A.W. & Olkin, I. (2007). Life distributions: Structure of nonparametric, semiparametric, and parametric families. New York, NY: Springer Science & Business Media, LLC.Google Scholar
30. Misra, N. & Francis, J. (2015). Relative ageing of (n−k+1)-out-of-n systems. Statistics and Probability Letters 106: 272–280.Google Scholar
31. Misra, N., Gupta, N., & Dhariyal, I.D. (2008). Preservation of some aging properties an stochastic orders by weighted distributions. Communications in Statistics—Theory and Methods 37: 627–644.Google Scholar
32. Misra, A.K. & Misra, N. (2012). Stochastic properties of conditionally independent mixture models. Journal of Statistical Planning and Inference 142: 1599–1607.Google Scholar
33. Mahfoud, M. & Patil, G.P. (1982). On weighted distributions. In: Kallianpur, G., Krishnaiah, P.R., Ghosh, J.K. (eds.), Statistics and probability: essays in honor of C.R. Rao, Amsterdam: North Holland, pp. 479–492.Google Scholar
34. Patil, G.P. & Rao, C.R. (1977). The weighted distributions: a survey and their applications. In: Krishnaiah, P.R. (ed.), Applications of statistics, Amsterdam: North Holland, pp. 383–405.Google Scholar
35. Pocock, S.J., Gore, S.M., & Kerr, G.R. (1982). Long-term survival analysis: the curability of breast cancer. Statistics in Medicine 1: 93–104.Google Scholar
36. Rao, C.R. (1965). On discrete distributions arising out of methods of ascertainment. In Patil, G.P. (ed.), Classical and contagious discrete distributions, Calcutta: Statistical Publishing Society, pp. 320–332.Google Scholar
37. Ross, S.M. (1983). Stochastic processes. New York: John Wiley & Sons.Google Scholar
38. Rezaei, M., Gholizadeh, B., & Izadkhah, S. (2015). On relative reversed hazard rate order. Communications in Statistics—Theory and Methods 44: 300–308.CrossRefGoogle Scholar
39. Sengupta, D. (1994). Another look at the moment bounds on reliability. Journal of Applied Probability 31: 777–787.Google Scholar
40. Sengupta, D. & Deshpande, J.V. (1994). Some results on the relative ageing of two life distributions. Journal of Applied Probability 31: 991–1003.CrossRefGoogle Scholar
41. Shaked, M. & Shanthikumar, J. G. (2007). Stochastic orders. New York, NY: Springer Science & Business Media, LLC.Google Scholar
42. Stoyan, D. (1983). Comparison methods for queues and other stochastic models. New York: Wiley.Google Scholar
43. Whitt, W. (1985). Uniform conditional variability ordering of probability distributions. Journal of Applied Probability 22: 619–633.Google Scholar
44. Zelen, M. & Feinleib, M. (1969). On the theory of screening for chronic diseases. Biometrika 56: 601–614.CrossRefGoogle Scholar