Abstract
In this article we study stochastic monotone properties of the deficit at ruin in terms of the increasing convex (concave) order. Also, we conduct comparisons on the extended deficit at ruin in the sense of the usual stochastic order and expectation. Additionally, the increasing convex (concave) order between the deficit at ruin and the amount of every drop in surplus is presented as well.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abouammoh A.M., Ahmad R. and Khalique A. (2000). On new renewal better than used classes of life distributions. Stat. Probab. Lett. 48, 189–194.
Bhattacharjee M.C., Ravi S., Vasudeva R. and Mohan N.R. (2003). New order preserving properties of geometric compounds. Stat. Probab. Lett. 64, 113–120.
Brémaud P. (2014). Fourier Analysis and Stochastic Processes. Springer, Switzerland.
Dufresne F. and Gerber H.U. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insur. Math. Econ. 10, 51–59.
Kaas R., Goovaerts M., Dhaene J. and Denuit M. (2008). Modern Actuarial Risk Theory. Springer, Heidelberg.
Kalashnikov V. (1997). Geometric Sums: Bounds for Rare Events with Applications. Kluwer Academic Publishers, Dordrecht.
Li H. and Li X. (2013). Stochastic Orders in Reliability and Risk. Springer, New York.
Lin X.S. and Willmot G.E. (1999). Analysis of a defective renewal equation arising in ruin theory. Insur. Math. Econ. 25, 63–84.
Marshall A.W. and Olkin I. (2007). Life Distributions. Springer, New York.
Psarrakos G. (2009). A note on convolutions of compound geometric distributions. Stat. Probab. Lett. 79, 1231–1237.
Psarrakos G. (2010). On the DFR property of the compound geometric distribution with applications in risk theory. Insur. Math. Econ. 47, 428–433.
Ross S.M. (1996). Stochastic Processes. Wiley, New York.
Shaked M. and Shanthikumar J.G. (2007). Stochastic Orders. Springer, New York.
Van Hoorn M. (1984). Algorithms and Approximations for Queueing Systems. CWI Tract 8, Amsterdam.
Willmot G.E. (2002). Compound geometric residual lifetime distributions and the deficit at ruin. Insur. Math. Econ. 30, 421–438.
Willmot G.E. and Cai J. (2004). On applications of residual lifetimes of compound geometric convolutions. J. Appl. Probab. 41, 802–815.
Willmot G.E. and Lin X.S. (2001). Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, New York.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the first two authors is supported by National Natural Science Foundation of China (11171278).
Rights and permissions
About this article
Cite this article
Li, C., Fang, R. & Li, X. Some Aging Properties Involved with Compound Geometric Distributions. Sankhya A 77, 337–350 (2015). https://doi.org/10.1007/s13171-015-0066-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-015-0066-7