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MEAN RESIDUAL LIFE FUNCTION FOR ADDITIVE AND MULTIPLICATIVE HAZARD RATE MODELS

Published online by Cambridge University Press:  15 December 2015

Ramesh C. Gupta
Ramesh C. Gupta*
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, Maine, 04469-5752, USA E-mail: rcgupta@maine.edu
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Abstract

This paper deals with the mean residual life function (MRLF) and its monotonicity in the case of additive and multiplicative hazard rate models. It is shown that additive (multiplicative) hazard rate does not imply reduced (proportional) MRLF and vice versa. Necessary and sufficient conditions are obtained for the two models to hold simultaneously. In the case of non-monotonic failure rates, the location of the turning points of the MRLF is investigated in both the cases. The case of random additive and multiplicative hazard rate is also studied. The monotonicity of the mean residual life is studied along with the location of the turning points. Examples are provided to illustrate the results.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 2 , April 2016 , pp. 281 - 297
Copyright
Copyright © Cambridge University Press 2015 

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References

1.Bebbington, M., Lai, C.D. & Zitikis, R. (2006). Useful periods for life time distributions with bathtub shaper hazard rate functions. IEEE Transactions on Reliability 55(2): 245–251.CrossRefGoogle Scholar
2.Bebbington, M., Lai, C.D. & Zitikis, R. (2007). Bathtub-type curves in reliability and beyond. Australian and New Zealand Journal of Statistics 49(3): 251–265.CrossRefGoogle Scholar
3.Bebbington, M., Lai, C.D. & Zitikis, R. (2008). Reduction in mean residual life in the presence of competing risk. Applied Stochastic Models in Business and Industry 24: 51–63.Google Scholar
4.Block, H.W., Jong, Y.K. & Savits, T.H. (1999). Bathtub functions and burn in. Probability in the Engineering and Informational Sciences 13: 497–507.CrossRefGoogle Scholar
5.Block, H.W. & Savits, T.H. (1997). Burn-in. Statistical Science 12: 1–19.Google Scholar
6.Block, H.W., Savits, T.H. & Singh, H. (2002). A criterion for burn-in that balances mean residual life and residual variance. Operations Research 50: 290–296.Google Scholar
7.Bryson, M.C. & Siddiqui, M.M. (1969). Some criteria for aging. Journal of the American Statistical Association 64: 1472–1483.CrossRefGoogle Scholar
8.Das, S. & Nanda, A.K. (2013). Some stochastic orders of dynamic additive mean residual life model. Journal of Statistical Planning and Inference 143: 400–407.Google Scholar
9.Duchateau, L. & Janssen, P. (2008). The Frailty Model. New York: Springer-Verlag.Google Scholar
10.Guess, F. & Proschan, F. (1988). Mean residual life: Theory and Applications. In Krishnaiah, P.R. & Rao, C.R. (eds.), Handbook of Statistics, Vol. 7, Amsterdam: Elsevier Science, pp. 215–224.Google Scholar
11.Gupta, R.C. & Akman, O. (1995). Mean residual life function for certain types of non-monotonic ageing. Stochastic Models 11(1): 219–225.Google Scholar
12.Gupta, R.C. & Kirmani, S.N.U.A. (1998). On the proportional mean residual life model and its implications. Statistics 32: 175–187.Google Scholar
13.Gupta, R.C. & Warren, R. (2001). Determination of change points of non-monotonic failure rates. Communications in Statistics 30(8&9): 1903–1920.Google Scholar
14.Gurland, J. & Sethuraman, J. (1995). How pooling failure data may reverse increasing failure rate. Journal of the American Statistical Association 90: 1416–1423.Google Scholar
15.Lai, C.D. & Xie, M. (2006). Stochastic ageing and dependence for reliability. New York: Springer-Verlag.Google Scholar
16.Lin, D.Y. & Ying, Z. (1994). Semiparametric analysis of the additive risk model. Biometrika 81: 61–71.CrossRefGoogle Scholar
17.Lin, D.Y., Oakes, D. & Ying, Z. (1998). Additive hazards regression with current status data. Biometrika 85: 289–298.Google Scholar
18.Lynch, J.D. (1999). On conditions for mixtures of increasing failure rate distributions to have increasing failure rate. Probability in the Engineering and Informational Sciences 13: 33–36.Google Scholar
19.Lynn, N.L. & Singpurwalla, N.D. (1997). Comment on “Burn in” makes us feel good. Statistical Science 12: 13–19.Google Scholar
20.Martinsussen, T. & Scheike, T.H. (2002). Efficient estimation in additive hazards regression with current status data. Biometrika, 89: 649–658.Google Scholar
21.Muth, E.J. (1977). Reliability models with positive memory derived from the mean residual life function. In Tsokos, C.P. & Shimmi, I. (eds.), The Theory and Applications of Reliability, Vol. 2, New York: Academic Press, pp. 401–435.Google Scholar
22.Prekopa, A. (1973). On logarithmic concave measures and functions. Acta Scientiarium Mathematicarum 33: 335–343.Google Scholar