Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T21:43:47.770Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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