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SHARP BOUNDS FOR SURVIVAL PROBABILITY WHEN AGEING IS NOT MONOTONE

Published online by Cambridge University Press:  06 April 2018

Ruhul Ali Khan  and
Murari Mitra
Ruhul Ali Khan
Affiliation:
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O. - Botanic Garden, Howrah-711103, West Bengal, India E-mail: ruhulali.khan@gmail.com
Murari Mitra
Affiliation:
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O. - Botanic Garden, Howrah-711103, West Bengal, India E-mails: murarimitra@yahoo.com, mmitra@math.iiests.ac.in
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Abstract

We exploit a novel bounding argument to obtain sharp bounds for survival functions belonging to the Increasing initially then Decreasing Mean Residual Life (IDMRL) class introduced by Guess, Hollander and Proschan (1986) [8]. The bounds obtained are in terms of the mean, change point and pinnacle of the mean residual life function. The bounds for the monotonic ageing classes Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) are obtained as special cases. Discussions on the bounds as well as two concrete illustrative examples are included.

Keywords

Life Distribution and Survival Function and Sharp Bounds and MRL Function and IDMRL Class and Change Point and Pinnacle
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 2 , April 2019 , pp. 205 - 219
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Aly, E.E.A.A. (1990). Tests for monotonicity properties of the mean residual life function. Scandinavian Journal of Statistics 17: 189–200.Google Scholar
2.Anis, M.Z. (2012). On some properties of the IDMRL class of life distributions. Journal of Statistical Planning and Inference 142: 3047–3055.Google Scholar
3.Barlow, , Richard, E. & Proschan, F. (1981). Statistical Theory of Reliability and Life Testing: Probability Models. To Begin With, MD, USA: Silver Spring.Google Scholar
4.Cox, D.R. (1962). Renewal Theory. London: Methuen.Google Scholar
5.Gertsbakh, I.B. (1989). Statistical Reliability Theory. New York: Marcel Dekker.Google Scholar
6.Glaser, R.E. (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association 75: 667–672.Google Scholar
7.Guess, F. & Proschan, F. (1985) Mean Residual Life: Theory and Applications. FSU Statistics Report M4702, AFOSR Technical Report No. 85–178.Google Scholar
8.Guess, F., Hollander, M. & Proschan, F. (1986). Testing exponentiality versus a trend change in mean residual life. The Annals of Statistics 14(4): 1388–1398.Google Scholar
9.Gupta, A.K., Zeng, W.B. & Wu, Y. (2010). Probability and Statistical Models: Foundations for Problems in Reliability and Financial Mathematics. New York: Springer.Google Scholar
10.Haines, A.L. & Singpurwalla, N.D. (1974). Some contributions to the stochastic characterization of wear. In Proschan, F., Serfling, R.J. (eds.), Reliability and Biometry. Society for Industrial and Applied Mathematics. Philadelphia, pp. 47–80.Google Scholar
11.Hawkins, D.L., Kochar, S. & Loader, C. (1992). Testing exponentiality against IDMRL distributions with unknown change point. The Annals of Statistics 20: 280–290.Google Scholar
12.Klefsjö, B. (1982). The HNBUE and HNWUE classes of life distributions. Naval Research Logistics Quarterly 29(2): 331–345.Google Scholar
13.Korzeniowski, A. & Opawski, A. (1976). Bounds for reliability in the NBU, NWU, NBUE and NWUE classes. Zastosowanià Matematyki Applicationes Mathematicae 15: 1–5.Google Scholar
14.Lai, C.D. & Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. New York: Springer.Google Scholar
15.Lim, J.H. & Park, D.H. (1998). A family of tests for trend change in mean residual life. Communications in Statistics–Theory and Methods 27: 1163–1179.Google Scholar
16.Marshall, A.W. & Proschan, F. (1972). Classes of distributions applicable in replacement with renewal theory implications. In Cam, Le (eds.), Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Vol. 11, pp. 395–415.Google Scholar
17.Meilijson, I. (1972). Limiting properties for the mean residual lifetime function. The Annals of Statistics 1: 354–357.Google Scholar
18.Mitra, M. & Basu, S.K. (1994). On a nonparametric family of life distributions and its dual. Journal of Statistical Planning and Inference 39: 385–397.Google Scholar
19.Mitra, M. & Basu, S.K. (1995). Change point estimation in non-monotonic aging models. Annals of the Institute of Statistical Mathematics 47: 483–491.Google Scholar
20.Na, M.H. & Lee, S. (2003). A family of IDMRL tests with unknown turning point. Statistics 37: 457–462.Google Scholar
21.Rajarshi, M.B. & Rajarshi, S.M. (1988). Bathtub distributions: A review. Communications in Statistics–Theory and Methods 17: 2597–2622.Google Scholar
22.Sengupta, D. (1994). Another look at the moment bounds on reliability. Journal of Applied Probability 31: 777–787.Google Scholar
23.Sengupta, D. & Das, S. (2016). Sharp bounds on DMRL and IMRL classes of life distributions with specified mean. Statistics and Probability Letters 119: 101–107.Google Scholar