Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-22T09:16:21.007Z Has data issue: false hasContentIssue false
  • English
  • Français

A class of location-independent variability orders, with applications

Part of: Distribution theory - Probability Mathematical economics Applications

Published online by Cambridge University Press:  14 July 2016

Moshe Shaked ,
Miguel A. Sordo  and
Alfonso Suárez-Llorens
Moshe Shaked*
Affiliation:
University of Arizona
Miguel A. Sordo*
Affiliation:
Universidad de Cádiz
Alfonso Suárez-Llorens*
Affiliation:
Universidad de Cádiz
*
∗Postal address: Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA. Email address: shaked@math.arizona.edu
∗∗Postal address: Departamento de Estadística e I. O., Universidad de Cádiz, C/ Duque de Nájera 8, CP: 11002, Cádiz, Spain.
∗∗Postal address: Departamento de Estadística e I. O., Universidad de Cádiz, C/ Duque de Nájera 8, CP: 11002, Cádiz, Spain.
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.

Li and Shaked (2007) introduced the family of generalized total time on test transform (TTT) stochastic orders, which is parameterized by a real function h that can be used to capture the preferences of a decision maker. It is natural to look for properties of these orders when there is an uncertainty in determining the appropriate function h. In this paper we study these orders when h is nondecreasing. We note that all these orders are location independent, and we characterize the dispersive order, and the location-independent riskier order, by means of the generalized TTT orders with nondecreasing h. Further properties, which strengthen known properties of the dispersive order, are given. A useful nontrivial closure property of the generalized TTT orders with nondecreasing h is obtained. Applications in poverty comparisons, risk management, and reliability theory are described.

Keywords

Dispersive orderexcess wealth orderstochastic orderlocation-independent riskier orderincreasing convex and concave orderspoverty measurerisk comparisontaxation policyNBUIFR

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 62P20: Applications to economics 91B30: Risk theory, insurance
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 407 - 425
Copyright
Copyright © Applied Probability Trust 2010 

References

Aebi, M., Embrechts, P. and Mikosch, T. (1992). A large claim index. Bull. Assoc. Swiss Actuaries 1992, 143–156.Google Scholar
Artzner, P. (1999). Application of coherent risk measures to capital requirements in insurance. N. Amer. Actuarial J. 3, 11–25.Google Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Belzunce, F. (1999). On a characterization of right spread order by the increasing convex order. Statist. Prob. Lett. 45, 103–110.Google Scholar
Belzunce, F., Pellerey, F., Ruiz, J. M. and Shaked, M. (1997). The dilation order, the dispersion order, and orderings of residual lives. Statist. Prob. Lett. 33, 263–275.Google Scholar
Chong, K. M. (1974). Some extensions of a theorem of Hardy, Littlewood and Pólya and their applications. Canad. J. Math. 26, 1321–1340.Google Scholar
De Giorgi, E. (2005). Reward-risk portfolio selection and stochastic dominance. J. Banking Finance 29, 895–926.Google Scholar
Duclos, J.-V. and Araar, A. (2006). Poverty and Equity: Measurement, Policy and Estimation with DAD. Springer, New York.Google Scholar
Duclos, J.-V. and Grégoire, P. (2002). Absolute and relative deprivation and the measurement of poverty. Rev. Income Wealth 48, 471–492.Google Scholar
Fagiuoli, E., Pellerey, F. and Shaked, M. (1999). A characterization of the dilation order and its applications. Statist. Papers 40, 393–406.Google Scholar
Fernandez-Ponce, J. M., Kochar, S. C. and Muñoz-Pérez, J. (1998). Partial orderings of distributions based on right-spread functions. J. Appl. Prob. 35, 221–228.Google Scholar
Föllmer, H. and Schied, A. (2004). Stochastic Finance, 2nd edn. Walter de Gruyer, Berlin.Google Scholar
Foster, J. E. and Shorrocks, A. F. (1988). Poverty orderings. Econometrica 56, 173–177.CrossRefGoogle Scholar
Foster, J., Greer, J. and Thorbecke, E. (1984). A class of decomposable poverty measures. Econometrica 52, 761–766.Google Scholar
Hagenaars, A. (1987). A class of poverty indices. Internat. Econom. Rev. 28, 583–607.Google Scholar
Hagenaars, A. J. M. and van Praag, B. M. S. (1985). A synthesis of poverty line definitions. Rev. Income Wealth 31, 139–154.Google Scholar
Jenkins, S. P. and Lambert, P. J. (1997). Three I's of poverty curves, with an analysis of UK poverty trends. Oxford Econom. Papers 49, 317–327.Google Scholar
Jewitt, I. (1989). Choosing between risky prospects: the characterization of comparative statics results, and location independent risk. Manag. Sci. 35, 60–70.Google Scholar
Jones, B. L. and Zitikis, R. (2003). Empirical estimation of risk measures and related quantities. N. Amer. Actuarial J. 7, 44–54.Google Scholar
Kayid, M. (2007). A general family of NBU classes of life distributions. Statist. Meth. 4, 185–195.Google Scholar
Kochar, S. C., Li, X. and Shaked, M. (2002). The total time on test transform and the excess wealth stochastic orders of distributions. Adv. Appl. Prob. 34, 826–845.Google Scholar
Li, X. and Shaked, M. (2007). A general family of univariate stochastic orders. J. Statist. Planning Infer. 137, 3601–3610.Google Scholar
Muñoz-Pérez, J. (1990). Dispersive ordering by the spread function. Statist. Prob. Lett. 10, 407–410.Google Scholar
Pellerey, F. and Shaked, M. (1997). Characterizations of the IFR and DFR aging notions by means of the dispersive order. Statist. Prob. Lett. 33, 389–393.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Shorrocks, A. F. (1995). Revisiting the Sen poverty index. Econometrica 63, 1225–1230.Google Scholar
Sordo, M. A. (2009). On the relationship of location-independent riskier order to the usual stochastic order. Statist. Prob. Lett. 79, 155–157.Google Scholar
Sordo, M. A. and Ramos, H. M. (2007). Characterizations of stochastic orders by L-functionals. Statist. Papers 48, 249–263.Google Scholar
Sordo, M. A., Ramos, H. M. and Ramos, C. D. (2007). Poverty measures and poverty orderings. SORT 31, 169–180.Google Scholar
Thon, D. (1983). A poverty measure. Indian Econom. J. 30, 55–70.Google Scholar
Wang, S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bull. 26, 74–92.Google Scholar
Wang, S. (1998). An actuarial index of the right-tail risk. N. Amer. Actuarial J. 2, 88–101.Google Scholar
Wang, S. S. and Young, V. R. (1998). Ordering risks: expected utility theory versus Yaari's dual theory of risk. Insurance Math. Econom. 22, 145–161.Google Scholar
Zheng, B. (2001). Statistical inference for poverty measures with relative poverty lines. J. Econometrics 101, 337–356.Google Scholar