Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T10:02:53.840Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME EXTENSIONS OF THE RESIDUAL LIFETIME AND ITS CONNECTION TO THE CUMULATIVE RESIDUAL ENTROPY

Published online by Cambridge University Press:  17 November 2011

Stella Kapodistria  and
Georgios Psarrakos
Stella Kapodistria
Affiliation:
Department of Mathematics and Computer Science & Eurandom, Eindhoven University of Technology, The Netherlands E-mail: [email protected]
Georgios Psarrakos
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Athens, Greece E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article we present a sequence of random variables with weighted tail distribution functions, constructed based on the relevation transform. For this sequence, we prove several recursive formulas and connections to the residual entropy through the unifying framework of the Dickson–Hipp operator. We also give some numerical examples to evaluate our results.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 1 , January 2012 , pp. 129 - 146
Copyright
Copyright © Cambridge University Press 2011

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.Asadi, M. & Zohrevand, Y. (2007). On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137: 1931–1941.CrossRefGoogle Scholar
2.Barlow, R.E. & Proschan, F. (1981). Statistical theory of life testing: Probability models. Silver Springs, MD: To Begin With.Google Scholar
3.Baxter, L.A. (1982). Reliability applications of the relevation transform. Naval Research Logistic Quarterly 29: 323–330.CrossRefGoogle Scholar
4.Boutsikas, M.V. & Vaggelatou, E. (2002). On the distance between convex-ordered random variables. Advances in Applied Probability 34: 349–374.CrossRefGoogle Scholar
5.Cai, J., Feng, R., & Willmot, G.E. (2009). On the expectation of total discounted operating costs up to default and its applications. Advances in Applied Probability 41: 495–522.Google Scholar
6.Denuit, M. & Van Bellegem, S. (2001). On the stop-loss and total variation distances between random sums. Insurance: Mathematics and Economics 53: 153–165.Google Scholar
7.Dickson, D.C.M. & Hipp, C. (2001). On the time to ruin for Erlang(2) risk processes. Insurance: Mathematics and Economics 29: 333–344.Google Scholar
8.Ebrahimi, N. (1996). How to measure uncertainty in the residual life distributions. Sankhya A 58: 48–57.Google Scholar
9.Feng, R. (2011). An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models. Insurance: Mathematics and Economics 48: 304–313.Google Scholar
10.Krakowski, M. (1973). The relevation transform and a generalization of the Gamma distribution function. Revue Francaise d'Automatique, Informatigue et Recherche Operationnelle 7(Ser. V-2): 107–120.CrossRefGoogle Scholar
11.Lau, K.S. & Rao, B.L.S.P. (1990). Characterization of the exponential distribution by the relevation transform. Journal of Applied Probability 27: 726–729.CrossRefGoogle Scholar
12.Li, S. & Garrido, J. (2004). On ruin for the Erlang(n) risk process. Insurance: Mathematics and Economics 34: 391–408.Google Scholar
13.Li, X. & Zhang, S. (2011). Some new results on Rényi entropi of residual life and inactivity time. Probability in the Engineering and Informational Sciences 25: 237–250.Google Scholar
14.Navarro, J. and Burkschat, M. (2011). Coherent systems based on sequential order statistics. Naval Research Logistics 58(2): 123–135.CrossRefGoogle Scholar
15.Navarro, J., del Aguila, Y., & Asadi, M. (2010). Some new results on the cumulative residual entropy. Journal of Statistical Planning and Inference 140: 310–322.Google Scholar
16.Oluyede, B.O. (2011). Exponential dominance and uncertainty for weighted residual life measures. Probability in the Engineering and Informational Sciences 25: 251–263.CrossRefGoogle Scholar
17.Patil, G.P. & Rao, C.R. (1977). Weighted distributions: A survey of their applications. In Applications of Statistics (Krishnaiah, P.R., Ed.), Amstredam: North-Holland, pp. 383–405.Google Scholar
18.Rao, M. (2005). More on a new concept of entropy and information. Journal of Theoretical Probability 18: 967–981.Google Scholar
19.Rao, M., Chen, Y., Vemuri, B.C., & Wang, F. (2004). Cumulative residual entropy: A new measure of information. IEEE Transactions on Information Theory 50: 1220–1228.CrossRefGoogle Scholar
20.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar
21.Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal 27: 379–423.Google Scholar
22.Shanthikumar, J.G. & Baxter, L.A. (1985). Closure properties of the relevation transform. Naval Research Logistic Quarterly 32: 185–189.CrossRefGoogle Scholar
23.Willmot, G.E. & Lin, X.S. (2001). Lundberg approximations for compound distributions with insurance applications, New York: Springer.CrossRefGoogle Scholar