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A class of location-independent variability orders, with applications

Published online by Cambridge University Press:  14 July 2016

Moshe Shaked
Affiliation:
University of Arizona
Miguel A. Sordo
Affiliation:
Universidad de Cádiz
Alfonso Suárez-Llorens
Affiliation:
Universidad de Cádiz
Corresponding
E-mail address:
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Abstract

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.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2010 

References

Aebi, M., Embrechts, P. and Mikosch, T. (1992). A large claim index. Bull. Assoc. Swiss Actuaries 1992, 143156.Google Scholar
Artzner, P. (1999). Application of coherent risk measures to capital requirements in insurance. N. Amer. Actuarial J. 3, 1125.CrossRefGoogle 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, 103110.CrossRefGoogle 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, 263275.CrossRefGoogle Scholar
Chong, K. M. (1974). Some extensions of a theorem of Hardy, Littlewood and Pólya and their applications. Canad. J. Math. 26, 13211340.CrossRefGoogle Scholar
De Giorgi, E. (2005). Reward-risk portfolio selection and stochastic dominance. J. Banking Finance 29, 895926.CrossRefGoogle 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, 471492.CrossRefGoogle Scholar
Fagiuoli, E., Pellerey, F. and Shaked, M. (1999). A characterization of the dilation order and its applications. Statist. Papers 40, 393406.CrossRefGoogle 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, 221228.CrossRefGoogle Scholar
Föllmer, H. and Schied, A. (2004). Stochastic Finance, 2nd edn. Walter de Gruyer, Berlin.CrossRefGoogle Scholar
Foster, J. E. and Shorrocks, A. F. (1988). Poverty orderings. Econometrica 56, 173177.CrossRefGoogle Scholar
Foster, J., Greer, J. and Thorbecke, E. (1984). A class of decomposable poverty measures. Econometrica 52, 761766.CrossRefGoogle Scholar
Hagenaars, A. (1987). A class of poverty indices. Internat. Econom. Rev. 28, 583607.CrossRefGoogle Scholar
Hagenaars, A. J. M. and van Praag, B. M. S. (1985). A synthesis of poverty line definitions. Rev. Income Wealth 31, 139154.CrossRefGoogle 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, 317327.CrossRefGoogle Scholar
Jewitt, I. (1989). Choosing between risky prospects: the characterization of comparative statics results, and location independent risk. Manag. Sci. 35, 6070.CrossRefGoogle Scholar
Jones, B. L. and Zitikis, R. (2003). Empirical estimation of risk measures and related quantities. N. Amer. Actuarial J. 7, 4454.CrossRefGoogle Scholar
Kayid, M. (2007). A general family of NBU classes of life distributions. Statist. Meth. 4, 185195.CrossRefGoogle 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, 826845.CrossRefGoogle Scholar
Li, X. and Shaked, M. (2007). A general family of univariate stochastic orders. J. Statist. Planning Infer. 137, 36013610.CrossRefGoogle Scholar
Muñoz-Pérez, J. (1990). Dispersive ordering by the spread function. Statist. Prob. Lett. 10, 407410.CrossRefGoogle 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, 389393.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Shorrocks, A. F. (1995). Revisiting the Sen poverty index. Econometrica 63, 12251230.CrossRefGoogle Scholar
Sordo, M. A. (2009). On the relationship of location-independent riskier order to the usual stochastic order. Statist. Prob. Lett. 79, 155157.CrossRefGoogle Scholar
Sordo, M. A. and Ramos, H. M. (2007). Characterizations of stochastic orders by L-functionals. Statist. Papers 48, 249263.CrossRefGoogle Scholar
Sordo, M. A., Ramos, H. M. and Ramos, C. D. (2007). Poverty measures and poverty orderings. SORT 31, 169180.Google Scholar
Thon, D. (1983). A poverty measure. Indian Econom. J. 30, 5570.Google Scholar
Wang, S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bull. 26, 7492.CrossRefGoogle Scholar
Wang, S. (1998). An actuarial index of the right-tail risk. N. Amer. Actuarial J. 2, 88101.CrossRefGoogle 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, 145161.CrossRefGoogle Scholar
Zheng, B. (2001). Statistical inference for poverty measures with relative poverty lines. J. Econometrics 101, 337356.CrossRefGoogle Scholar
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