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On an ageing class based on the moment generating function order

Part of: Special processes Survival analysis and censored data

Published online by Cambridge University Press:  26 July 2018

Shyamal Ghosh  and
Murari Mitra
Shyamal Ghosh*
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, P.O. Botanic Garden, Howrah, 711103, West Bengal, India.
* Postal address: Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O. Botanic Garden, Howrah, 711103, West Bengal, India.
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Abstract

We develop shock model theory in different scenarios for the ℳ-class of life distributions introduced by Klar and Müller (2003). We also study the cumulative damage model of A-Hameed and Proschan (1975) in the context of ℳ-class and establish analogous results. We obtain moment bounds and explore weak convergence issues within the ℳ-class of life distributions.

Keywords

Moment generating function orderNHPPnonstationary pure-birth processcumulative damage modelweak convergencemoment convergence

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 402 - 415
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Abouammoh, A. M., Hindi, M. I. and Ahmed, A. N. (1988). Shock models with NBUFR and NBAFR survivals. Trabajos Estadistica 3, 97–113. Google Scholar
[2]A-Hameed, M. S. and Proschan, F. (1973). Nonstationary shock models. Stoch. Process. Appl. 1, 383–404. Google Scholar
[3]A-Hameed, M. S. and Proschan, F. (1975). Shock models with underlying birth process. J. Appl. Prob. 12, 18–28. Google Scholar
[4]Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York. Google Scholar
[5]Basu, S. K. and Bhattacharjee, M. C. (1984). On weak convergence within the HNBUE family of life distributions. J. Appl. Prob. 21, 654–660. Google Scholar
[6]Basu, S. K. and Simons, G. (1983). Moment spaces for IFR distributions, applications and related materials. In Contributions to Statistics, North Holland, Amsterdam, pp. 27–46. Google Scholar
[7]Bhattacharyya, G. and Fries, A. (1982). Fatigue failure models: Birnbaum-Saunders vs. inverse Gaussian. IEEE Trans. Reliab. 31, 439–441. Google Scholar
[8]Billingsley, P. (1979). Probability and Measure. John Wiley, New York. Google Scholar
[9]Block, H. W. and Savits, T. H. (1978). Shock models with NBUE survival. J. Appl. Prob. 15, 621–628. Google Scholar
[10]Chaudhuri, G. (1995). A note on the L-class of life distributions. Sankhyā A 57, 158–160. Google Scholar
[11]Esary, J. D., Marshall, A. W. and Proschan, F. (1973). Shock models and wear processes. Ann. Prob. 1, 627–649. Google Scholar
[12]Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York. Google Scholar
[13]Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952). Inequalities, 2nd edn. Cambridge University Press. Google Scholar
[14]Klar, B. (2002). A note on the ℒ-class of life distributions. J. Appl. Prob. 39, 11–19. Google Scholar
[15]Klar, B. (2005). Tests for exponentiality against the ℳ and ℒ$ℳ-classes of life distributions. Test 14, 543–565. Google Scholar
[16]Klar, B. and Müller, A. (2003). Characterizations of classes of lifetime distributions generalizing the NBUE class. J. Appl. Prob. 40, 20–32. Google Scholar
[17]Klefsjö, B. (1981). HNBUE survival under some shock models. Scand. J. Statist. 8, 39–47. Google Scholar
[18]Klefsjö, B. (1983). A useful ageing property based on the Laplace transform. J. Appl. Prob. 20, 615–626. Google Scholar
[19]Kochar, S. C. and Deshpande, J. V. (1985). On exponential scores statistic for testing against positive ageing. Statist. Prob. Lett. 3, 71–73. Google Scholar
[20]Kozakiewicz, W. (1947). On the convergence of sequences of moment generating functions. Ann. Math. Statist. 18, 61–69. Google Scholar
[21]Lai, C.-D. and Xie, M. (2006). Stochastic Ageing and Dependence for Reliability. Springer, New York. Google Scholar
[22]Li, X. (2004). Some properties of ageing notions based on the moment-generating-function order. J. Appl. Prob. 41, 927–934. Google Scholar
[23]Loève, M. (1963). Probability Theory, 3rd edn. Van Nostrand, Princeton, NJ. Google Scholar
[24]Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York. Google Scholar
[25]Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester. Google Scholar
[26]Nanda, A. K. and Paul, P. (2004). An observation from DMRL life distribution. Statist. Meth. 6, 57–65, 235. Google Scholar
[27]Pellerey, F. (1994). Shock models with underlying counting process. J. Appl. Prob. 31, 156–166. Google Scholar
[28]Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York. Google Scholar
[29]Zhang, S. and Li, X. (2010). Some new results on the moment generating function order and related life distributions. J. Appl. Prob. 47, 923–933. Google Scholar