Mean residual life function, associated orderings and properties

IEEE Transactions on Reliability

Volumn 59, Issue 1, 2010, Pages 55-65

  Nanda, Asok K ^a   Bhattacharjee, Subarna ^b   Balakrishnan, Narayanaswamy ^c  

^a INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH KOLKATA   (India)

^b KIIT UNIVERSITY   (India)

^c MCMASTER UNIVERSITY   (Canada)

Author keywords

Hazard rate order; Order statistics; Stochastic order

Indexed keywords

COMPONENT LEVELS; FAILURE RATE FUNCTIONS; HAZARD RATE ORDER; MEAN RESIDUAL LIFE; MEAN RESIDUAL LIFE FUNCTIONS; ORDER STATISTICS; PARTIAL ORDERING; REPLACEMENT STRATEGY; RESIDUAL LIFE; SHOCK MODELS; STOCHASTIC ORDER; SYSTEM LEVELS;

CHARACTERIZATION;

STOCHASTIC SYSTEMS;

EID: 77949268269     PISSN: 00189529     EISSN: None     Source Type: Journal    
DOI: 10.1109/TR.2009.2035791     Document Type: Article

References (70)

1. S. E. Abu-Youssef, "A moment inequality for decreasing (increasing) mean residual life distributions with hypothesis testing application," Statistics & Probability Letters, vol. 57, pp. 171-177, 2002.
2. R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models.,:, 1981, To Begin With, Silver Spring, MD.
3. J. Bartoszewicz, "Dispersive functions and stochastic orders," Applicationes Mathematicae, vol. 24, pp. 429-444, 1997.
4. L. Bekker and J. Mi, "Shape and crossing properties of mean residual life functions," Statistics & Probability Letters, vol. 64, pp. 225-234, 2003.
5. F. Belzunce, "On a characterization of right spread order by the increasing convex order," Statistics & Probability Letters, vol. 45, pp. 103-110, 1999.
6. F. Belzunce, J. Candel, and J. M. Ruiz, "Dispersive ordering and characterizations of ageing classes," Statistics & Probability Letters, vol. 28, pp. 321-327, 1996.
7. F. Belzunce, X. Gao, T. Hu, and F. Pellerey, "Characterizations of the hazard rate order and IFR aging notion," Statistics & Probability Letters, vol. 70, pp. 235-242, 2004.
8. F. Belzunce, F. Pellerey, J. M. Ruiz, and M. Shaked, "The dilation order, the dispersive order and orderings of residual lives," Statistics & Probability Letters, vol. 33, pp. 263-275, 1997.
9. M. C. Bhattacharjee, "The class of mean residual lives and some consequences," SIAM Journal on Algebraic and Discrete Methods, vol. 3, no. 1, pp. 56-65, 1982.
10. P. J. Boland and E. El-Neweihi, "Component redundancy vs. system redundancy in the hazard rate ordering," IEEE Trans. Reliability, vol. 44, pp. 614-619, 1995.
11. D. M. Bradley and R. C. Gupta, "Limiting behavior of the mean residual life," Annals of the Institute of Statistical Mathematics, vol. 55, pp. 217-226, 2003.
12. M. Brown and J. G. Shanthikumar, "Comparing the variability of random variables and point processes," Probability in the Engineering and Informational Sciences, vol. 12, pp. 425-444, 1998.
13. M. C. Bryson and M. M. Siddiqui, "Some criteria for aging," Journal of the American Statistical Association, vol. 64, pp. 1472-1483, 1969.
14. J. Cao and Y. Wang, "The NBUC and NWUC classes of life distributions," Journal of Applied Probability, vol. 28, pp. 473-479, 1991.
15. P. Capéraà, "Tail ordering and asymptotic efficiency of rank tests," Annals of Statistics, vol. 16, pp. 470-478, 1988.
16. Y. P. Chaubey and P. K. Sen, "Smooth isotonic estimation of density, hazard and MRL functions," Calcutta Statistical Association Bulletin, vol. 52, pp. 99-116, 2002.
17. D. R. Cox, "Regression models and life tables (with discussion)," Journal of the Royal Statistical Society, Series B, vol. 34, pp. 187-202, 1972.
18. A. Di Crescenzo, "A probabilistic analogue of the mean value theorem and its applications to reliability theory," Journal of Applied Probability, vol. 36, pp. 706-719, 1999.
19. A. Di Crescenzo and M. Longobardi, "The up reversed hazard rate stochastic order," Scientiae Mathematicae Japonicae, vol. 54, pp. 575-581, 2001.
20. N. Ebrahimi, "Estimation of two ordered mean residual lifetime functions," Biometrics, vol. 49, pp. 409-417, 1993.
21. E. Fagiuoli, F. Pellerey, and M. Shaked, "A characterization of the dilation order and its applications," Statistical Papers, vol. 40, pp. 393-406, 1999.
22. M. Finkelstein, "On the shape of the mean residual life function," Applied Stochastic Models in Business and Industry, vol. 18, pp. 135-146, 2002.
23. M. Finkelstein, "On relative ordering of mean residual lifetime functions," Statistics & Probability Letters, vol. 76, pp. 939-944, 2006.
24. E. Frostig, "On risk dependence and mrl ordering," Statistics & Probability Letters, vol. 76, pp. 231-243, 2006.
25. F. Guess, M. Hollander, and F. Proschan, "Testing exponentiality versus a trend change in mean residual life," Annals of Statistics, vol. 14, pp. 1388-1398, 1986.
26. F. Guess and F. Proschan, P. R. Krishnaiah and C. R. Rao, Eds., "Mean residual life: Theory and applications," in Handbook of Statistics. Amsterdam: Elsevier, 1988, vol. 7, pp. 215-224.
27. R. C. Gupta, "On the mean residual function in survival studies," in Statistical Distributions in Scientific Work, C. Taillie, G. P. Patil, and B. A. Baldessari, Eds.:, 1981, vol. 5, pp. 327-334, D. Reidel, Dordrecht.
28. R. C. Gupta and S. N. U. A. Kirmani, "On order relations between reliability measures," Communications in Statistics - Stochastic Models, vol. 3, pp. 149-156, 1987.
29. R. C. Gupta and S. N. U. A. Kirmani, "On the proportional mean residual life model and its implications," Statistics, vol. 32, pp. 175-187, 1998.
30. W. J. Hall and J. A. Wellner, M. Csörgö, D. A. Dawson, J. N. K. Rao, and A. K. Md. E. Saleh, Eds., "Mean residual life," in Statistics and Related Topics. Amsterdam: North-Holland, 1981, pp. 169-184.
31. T. Hu, A. Kundu, and A. K. Nanda, Y. Hayakawa, T. Irony, and M. Xie, Eds., "On generalized orderings and ageing properties with their implications," in System and Bayesian Reliability. Singapore: World Scientific, 2001, pp. 199-228.
32. T. Hu, A. K. Nanda, H. Xie, and Z. Zhu, "Properties of some stochastic orders: A unified study," Naval Research Logistics, vol. 51, pp. 193-216, 2004.
33. T. Hu and Y. Wei, "Stochastic comparisons of spacings from restricted families of distributions," Statistics & Probability Letters, vol. 53, pp. 91-99, 2001.
34. T. Hu, Z. Zhu, and Y. Wei, "Likelihood ratio and mean residual life orders for order statistics of heterogenous random variables," Probability in the Engineering and Informational Sciences, vol. 15, pp. 259-272, 2001.
35. I. Jewitt, "Choosing between risky prospects: The characterization of comparative statics results, and location independent risk," Management Science, vol. 35, pp. 60-70, 1989.
36. K. Joag-Dev, S. Kochar, and F. Proschan, "A general composition theorem and its applications to certain partial orderings of distributions," Statistics & Probability Letters, vol. 22, pp. 111-119, 1995.
37. S. Kapan and T. A. Mazzuchi, "Interrelationship of burn-in criteria," in YA/EM 2004-Y öneylem Arastirmasi/Endüstri Mühendislii-XXIV Ulusal Kongresi.:, 2004, pp. 15-18, Haziran, Gaziantep-Adana.
38. C. Kay and W. W. Yi, "Preservation of some partial orderings under nonhomogeneous Poisson shock model and Laplace transform," Microelectronics and Reliability, vol. 37, pp. 451-455, 1997.
39. S. C. Kochar, "Partial orderings of distributions based on right-spread functions," Journal of Applied Probability, vol. 35, pp. 221-228, 1998.
40. S. C. Kochar, H. Mukherjee, and F. J. Samaniego, "Estimation of monotone mean residual life," Annals of Statistics, vol. 28, pp. 905-921, 2000.
41. S. C. Kochar and D. D. Wiens, "Partial orderings of life distributions with respect to their ageing properties," Naval Research Logistics, vol. 34, pp. 823-829, 1987.
42. J. Kupka and S. Loo, "The hazard and vitality measures of ageing," Journal of Applied Probability, vol. 26, pp. 532-542, 1989.
43. C. D. Lai and M. Xie, Stochastic Ageing and Dependence for Reliability. New York: Springer-Verlag, 2006.
44. E. L. Lehmann, "The power of rank test," Annals of Mathematical Statistics, vol. 24, pp. 23-43, 1953.
45. R. E. Lillo, A. K. Nanda, and M. Shaked, N. Limnios and M. Nikulin, Eds., "Some shifted stochastic orders," in Recent Advances in Reliability Theory: Methodology, Practice and Inference. Boston: Birkhäuser, 2001, pp. 85-103.
46. J. H. Lim and D. H. Park, "A family of test statistics for DMRL (IMRL) alternatives," Journal of Nonparametric Statistics, vol. 8, pp. 293-305, 1997.
47. G. Maguluri and C. H. Zhang, "Estimation in the mean residual life regression model," Journal of the Royal Statistical Society, Series B, vol. 56, pp. 477-489, 1994.
48. I. Meilijson, "Limiting properties of the mean residual life function," Annals of Mathematical Statistics, vol. 43, pp. 354-357, 1972.
49. J. Mi, "Bathtub failure rate and upside-down bathtub mean residual life," IEEE Trans. Reliability, vol. 44, pp. 388-391, 1995.
50. J. Mi, "A general approach to the shape of failure rate and mrl functions," Naval Research Logistics, vol. 51, pp. 543-556, 2004.
51. E. J. Muth, C. P. Tsokos and I. N. Shimi, Eds., "Reliability models with positive memory derived from the mean residual life function," in Theory and Applications of Reliability. New York: Academic Press, 1977, pp. 401-435.
52. T. Nakai, "A partially observable decision problem under a shifted likelihood ratio ordering," Mathematical and Computer Modelling, vol. 22, pp. 237-246, 1995.
53. A. K. Nanda, "Stochastic orders in terms of Laplace transforms," Calcutta Statistical Association Bulletin, vol. 45, pp. 195-201, 1995.
54. A. K. Nanda, S. Bhattacharjee, and S. S. Alam, "Properties of proportional mean residual life model," Statistics & Probability Letters, vol. 76, pp. 880-890, 2006.
55. A. K. Nanda and K. Jain, "Some weighted distribution results on univariate and bivariate cases," Journal of Statistical Planning and Inference, vol. 77, pp. 169-180, 1999.
56. A. K. Nanda, K. Jain, and H. Singh, "On closure of some partial orderings under mixtures," Journal of Applied Probability, vol. 33, pp. 698-706, 1996.
57. A. K. Nanda and P. Paul, "An observation from DMRL life distribution," Statistical Methods, vol. 6, pp. 57-65, 2004,
58. [Errata: Statistical Methods (2004), vol. 6, 235].
59. D. Oakes and T. Dasu, "A note on residual life," Biometrika, vol. 77, pp. 409-410, 1990.
60. F. Pellerey, "On the preservation of some orderings of risks under convolution," Insurance: Mathematics and Economics, vol. 16, pp. 23-30, 1995.
61. D. Sengupta and J. V. Deshpande, "Some results on the relative aging of two life distributions," Journal of Applied Probability, vol. 31, pp. 991-1003, 1994.
62. M. Shaked, "Dispersive ordering of distributions," Journal of Applied Probability, vol. 19, pp. 310-320, 1982.
63. M. Shaked and J. G. Shanthikumar, Stochastic Orders. New York: Springer, 2007.
64. M. Shaked and T. Wong, "Preservation of stochastic orderings under random mapping by point processes," Probability in the Engineering and Informational Sciences, vol. 9, pp. 563-580, 1995.
65. J. G. Shanthikumar and D. D. Yao, "The preservation of likelihood ratio ordering under convolution," Stochastic Processes and Their Applications, vol. 23, pp. 259-267, 1986.
66. H. Singh, "On partial orderings of life distributions," Naval Research Logistics, vol. 36, pp. 103-110, 1989.
67. H. Singh and K. Jain, "Preservation of some partial orderings under Poisson shock models," Journal of Applied Probability, vol. 21, pp. 713-716, 1989.
68. H. Singh and R. S. Singh, "On allocation of spares at component level versus system level," Journal of Applied Probability, vol. 34, pp. 283-287, 1997.
69. K. L. Wong, "The physical basis for the roller-coaster hazard rate curve for electronics," Quality and Reliability Engineering International, vol. 7, pp. 489-495, 2007.
70. H. Zahedi, "Proportional mean residual life model," Journal of Statistical Planning and Inference, vol. 29, pp. 221-228, 1991.