Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T10:10:48.886Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE ELASTICITY OF EXPECTED INTEREPOCH INTERVALS IN A NON-HOMOGENEOUS POISSON PROCESS UNDER SMALL VARIATIONS OF HAZARD RATE

Published online by Cambridge University Press:  23 April 2019

Georgios Psarrakos  and
Abdolsaeed Toomaj
Georgios Psarrakos
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece E-mail: [email protected]
Abdolsaeed Toomaj
Affiliation:
Department of Mathematics and Statistics, Gonbad Kavous University, Gonbad Kavous, Iran E-mail: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The elasticity of life expectancy is an important feature in life tables. It is also known as life table entropy in the areas of demography and biology, and as normalized cumulative residual entropy in reliability theory. The elasticity of life expectancy provides useful information on studying the way in which small variations in the force of mortality (or hazard rate) affects the life expectancy. In this paper, a perturbation analysis of the hazard rate to the expected interepoch intervals in a non-homogeneous Poisson process is applied, and further interpretations are given by using a normalized version of the generalized cumulative residual entropy. Properties of the elasticity, including ordering results, bounds and empirical estimation, are obtained. Moreover, the dynamic version of the elasticity is studied, and some monotonicity and characterization results are given.

Keywords

elasticity of life expectancygeneralized cumulative residual entropyhazard ratemean residual lifetimeproportional hazards model
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 4 , October 2020 , pp. 550 - 569
Copyright
Copyright © Cambridge University Press 2019

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.Ascher, H. & Feingold, H. (1984). Repairable system reliability: modelling, inference, misconceptions and their causes. New York: Marcel Decker.Google Scholar
3.Barlow, R.E. & Proschan, F. (1975). Statistical theory of reliability and life testing. New York: Holt, Rinehart and Winston.Google Scholar
4.Baxter, L.A. (1982). Reliability applications of the relevation transform. Naval Research Logistics Quarterly 29: 323–330.CrossRefGoogle Scholar
5.Beichelt, F. (1993). A unifying treatment of replacement policies with minimal repair. Naval Research Logistics 40: 51–67.3.0.CO;2-V>CrossRefGoogle Scholar
6.Block, H.W., Borges, W.S. & Savits, T.H. (1985). Age-dependent minimal repair. Journal of Applied Probability 22: 370–385.CrossRefGoogle Scholar
7.Chamany, A. & Baratpour, S. (2014). A dynamic discrimination information based on cumulative residual entropy and its properties. Communications in Statistics-Theory and Methods 43(6): 1041–1049.CrossRefGoogle Scholar
8.Demetrius, L. (1976). Measures of variability in age-structured populations. Journal of Theoretical Biology 63: 397–404.CrossRefGoogle ScholarPubMed
9.Di Crescenzo, A. & Longobardi, M. (2006). On weighted residual and past entropies. Scientiae Mathematicae Japonicae 64: 255–266.Google Scholar
10.Di Crescenzo, A. & Longobardi, M. (2009). On cumulative entropies. Journal of Statistical Planning and Inference 139: 4072–4087.CrossRefGoogle Scholar
11.Di, Crescenzo A. & Toomaj, A. (2017). Further results on the generalized cumulative entropy. Kybernetika 53(5): 959–982.Google Scholar
12.Goldman, N. & Lord, G. (1986). A new look at entropy and the life table. Demography 23(2): 275–282.CrossRefGoogle Scholar
13.Gupta, R.C. & Kirmani, S.N.U.A. (2000). Residual coefficient of variation and some characterization results. Journal of Statistical Planning and Inference 91: 23–31.CrossRefGoogle Scholar
14.Hakkert, R. (1987). Lifetable transformations and inequality measures: some noteworthy formal relations. Demography 24: 615–622.CrossRefGoogle Scholar
15.Kapodistria, S. & Psarrakos, G. (2012). Some extensions of the residual lifetime and its connection to the cumulative residual entropy. Probability in the Engineering and Informational Sciences 26: 129–146.CrossRefGoogle Scholar
16.Karlin, S. (1968). Total positivity. Stanford, CA, USA: Stanford University Press.Google Scholar
17.Kayal, S. (2018). On weighted generalized cumulative residual entropy of order n. Methodology and Computing in Applied Probability 20: 487–503.CrossRefGoogle Scholar
18.Kayal, S. & Moharana, R. (2017). On weighted cumulative residual entropy. Journal of Statistics and Management Systems 20: 153–173.CrossRefGoogle Scholar
19.Kenett, R.S. & Pollak, M. (1996). Data-analytic aspects of the Shiryayev-Roberts control chart: surveillance of a non-homogeneous Poisson process. Journal of Applied Statistics 23: 125–137.CrossRefGoogle Scholar
20.Keyfitz, N. (1977). Applied mathematical demography. New York: Wiley.Google Scholar
21.Leser, C.E.V. (1955). Variations in mortality and life-expectation. Population Studies 9: 67–71.CrossRefGoogle Scholar
22.Mirali, M., Baratpour, S. & Fakoor, V. (2017). On weighted cumulative residual entropy. Communications in Statistics-Theory and Methods 46: 2857–2869.CrossRefGoogle Scholar
23.Misagh, F. & Yari, G.H. (2011). On weighted interval entropy. Statistics and Probability Letters 81: 188–194.CrossRefGoogle Scholar
24.Mitra, S. (1978). A short note on the Taeuber paradox. Demography 15(4): 621–623.CrossRefGoogle Scholar
25.Navarro, J. & Psarrakos, G. (2017). Characterizations based on generalized cumulative residual entropy functions. Communications in Statistics-Theory and Methods 46: 1247–1260.CrossRefGoogle Scholar
26.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.CrossRefGoogle Scholar
27.Pellerey, F., Shaked, M. & Zinn, J. (2000). Nonhomogeneous Poisson processes and logconcavity. Probability in the Engineering and Informational Sciences 14: 353–373.CrossRefGoogle Scholar
28.Psarrakos, G. & Navarro, J. (2013). Generalized cumulative residual entropy and record values. Metrika 27: 623–640.CrossRefGoogle Scholar
29.Psarrakos, G. & Toomaj, A. (2017). On the generalized cumulative residual entropy with applications in actuarial science. Journal of Computational and Applied Mathematics 309: 186–199.CrossRefGoogle Scholar
30.Rao, M. (2005). More on a new concept of entropy and information. Journal of Theoretical Probability 18: 967–981.CrossRefGoogle Scholar
31.Rao, M., Chen, Y., Vemuri, B. & Fei, W. (2004). Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50(6): 1220–1228.CrossRefGoogle Scholar
32.Shannon, C.E. (1948). A mathematical theory of communication. Bell System Technical Journal 27: 379–423 and 623–656.CrossRefGoogle Scholar
33.Sordo, M.A. & Psarrakos, G. (2017). Stochastic comparisons of interfailure times under a relevation replacement policy. Journal of Applied Probability 54: 134–145.CrossRefGoogle Scholar
34.Toomaj, A., Sunoj, S. & Navarro, J. (2017). Some properties of the cumulative residual entropy of coherent and mixed systems. Journal of Applied Probability 54: 379–393.CrossRefGoogle Scholar
35.Vaupel, J.W. (1986). How change in age-speciffc mortality affects life expectancy. Population Studies 40(1): 147–157.CrossRefGoogle ScholarPubMed
36.Wang, S. (1998). An actuarial index of the right-tail risk. North American Actuarial Journal 2: 88–101.CrossRefGoogle Scholar