Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-22T08:37:00.012Z Has data issue: false hasContentIssue false
  • English
  • Français

Extension of the past lifetime and its connection to the cumulative entropy

Part of: Survival analysis and censored data Distribution theory - Probability Sufficiency and information Communication, information

Published online by Cambridge University Press:  30 March 2016

Antonio Di Crescenzo  and
Abdolsaeed Toomaj
Antonio Di Crescenzo*
Affiliation:
Università degli Studi di Salerno
Abdolsaeed Toomaj*
Affiliation:
Gonbad Kavous University
*
∗Postal address: Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, I-84084 Fisciano (SA), Italy. Email address: adicrescenzo@unisa.it
∗∗Postal address: Department of Statistics, Gonbad Kavous University, Gonbad Kavous, Iran. Email address: ab.toomaj@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

Keywords

Relevation transformreversed relevation transformproportional hazard rate modelproportional reversed hazard rate modelweighted cumulative distributioncumulative entropy

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62B10: Information-theoretic topics 62N05: Reliability and life testing 94A17: Measures of information, entropy
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1156 - 1174
Copyright
Copyright © Applied Probability Trust 2015 

References

[1] Abate, J. and Whitt, W. (1996). An operational calculus for probability distributions via Laplace transforms. Adv. Appl. Prob. 28, 75–113.CrossRefGoogle Scholar
[2] Bartoszewicz, J. (2009). On a representation of weighted distributions. Statist. Prob. Lett. 79, 1690–1694.Google Scholar
[3] Baxter, L. A. (1982). Reliability applications of the relevation transform. Naval Res. Logistics Quart. 29, 321–330.CrossRefGoogle Scholar
[4] Block, H. W., Savits, T. H. and Singh, H. (1998). The reversed hazard rate function. Prob. Eng. Inf. Sci. 12, 69–90.CrossRefGoogle Scholar
[5] Bloom, S. (1997). First and second order Opial inequalities. Studia Math. 126, 27–50.CrossRefGoogle Scholar
[6] Burkschat, M. and Navarro, J. (2014). Asymptotic behavior of the hazard rate in systems based on sequential order statistics. Metrika 77, 965–994.Google Scholar
[7] Cai, J., Feng, R. and Willmot, G. E. (2009). On the expectation of total discounted operating costs up to default and its applications. Adv. Appl. Prob. 41, 495–522.Google Scholar
[8] Dickson, D. C. M. and Hipp, C. (2001). On the time to ruin for Erlang(2) risk process. Insurance Math. Econom. 29, 333–344.CrossRefGoogle Scholar
[9] Di Crescenzo, A. (1999). A probabilistic analogue of the mean value theorem and its applications to reliability theory. J. Appl. Prob. 36, 706–719.CrossRefGoogle Scholar
[10] Di Crescenzo, A. (2000). Some results on the proportional reversed hazards model. Statist. Prob. Lett. 50, 313–321.CrossRefGoogle Scholar
[11] Di Crescenzo, A. and Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. J. Appl. Prob. 39, 434–440.Google Scholar
[12] Di Crescenzo, A. and Longobardi, M. (2009). On cumulative entropies. J. Statist. Planning Inference 139, 4072–4087.Google Scholar
[13] Di Crescenzo, A., Martinucci, B. and Mulero, J. (2016). A quantile-based probabilistic mean value theorem. To appear in Prob. Eng. Inf. Sci. Google Scholar
[14] Dimitrov, B., Chukova, S. and Green, D. Jr. (1997). Probability distributions in periodic random environment and their applications. SIAM J. Appl. Math. 57, 501–517.Google Scholar
[15] Gupta, R. C., Gupta, P. L. and Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Commun. Statist. Theory Meth. 27, 887–904.Google Scholar
[16] Gupta, R. C., Kannan, N. and Raychaudhuri, A. (1997). Analysis of lognormal survival data. Math. Biosci. 139, 103–115.Google Scholar
[17] Gupta, R. D. and Nanda, A. K. (2001). Some results on reversed hazard rate ordering. Commun. Statist. Theory Meth. 30, 2447–2457.Google Scholar
[18] Kapodistria, S. and Psarrakos, G. (2012). Some extensions of the residual lifetime and its connection to the cumulative residual entropy. Prob. Eng. Inf. Sci. 26, 129–146.Google Scholar
[19] Kijima, M. and Ohnishi, M. (1999). Stochastic orders and their application in financial optimization. Math. Meth. Operat. Res. 50, 351–372.CrossRefGoogle Scholar
[20] Krakowski, M. (1973). The relevation transform and a generalization of the gamma distribution function. Rev. Française Automat. Informat. Recherche Operat. 7, 107–120.Google Scholar
[21] Lau, K.-S. and Prakasa Rao, B. L. S. (1990). Characterization of the exponential distribution by the relevation transform. J. Appl. Prob. 27, 726–729.CrossRefGoogle Scholar
[22] Lau, K.-S. and Prakasa Rao, B. L. S. (1992). Letter to the editor: Characterization of the exponential distribution by the relevation transform. J. Appl. Prob. 29, 1003–1004.Google Scholar
[23] Lehmann, E. L. (1953). The power of rank tests. Ann. Math. Statist. 24, 23–43.CrossRefGoogle Scholar
[24] Li, S. and Garrido, J. (2004). On ruin for the Erlang(n) risk process. Insurance Math. Econom. 34, 391–408.Google Scholar
[25] Li, Y., Yu, L. and Hu, T. (2012). Probability inequalities for weighted distributions. J. Statist. Planning Inference 142, 1272–1278.Google Scholar
[26] Mudholkar, G. S. and Hutson, A. D. (1996). The exponentiated Weibull family: some properties and a flood data application. Commun. Statist. Theory Meth. 25, 3059–3083.Google Scholar
[27] Mudholkar, G. S., Srivastava, D. K. and Freimer, M. (1995). The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37, 436–445.Google Scholar
[28] Muliere, P., Parmigiani, G. and Polson, N. G. (1993). A note on the residual entropy function. Prob. Eng. Inf. Sci. 7, 413–420.Google Scholar
[29] Navarro, J., Sunoj, S. M. and Linu, M. N. (2011). Characterizations of bivariate models using dynamic Kullback-Leibler discrimination measures. Statist. Prob. Lett. 81, 1594–1598.Google Scholar
[30] Psarrakos, G. and Navarro, J. (2013). Generalized cumulative residual entropy and record values. Metrika 76, 623–640.CrossRefGoogle Scholar
[31] Rezaei, M., Gholizadeh, B. and Izadkhah, S. (2015). On relative reversed hazard rate order. Commun. Statist. Theory Meth. 44, 300–308.Google Scholar
[32] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
[33] Shannon, C. E. (1948). A mathematical theory of communication. Bell. Syst. Tech. J. 27, 379–423, 623–656.Google Scholar