Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T20:37:51.527Z Has data issue: false hasContentIssue false
  • English
  • Français

On sufficient conditions for the comparison in the excess wealth order and spacings

Published online by Cambridge University Press:  24 March 2016

Félix Belzunce ,
Carolina Martínez-Riquelme ,
José M. Ruiz  and
Miguel A. Sordo
Félix Belzunce*
Affiliation:
Facultad de Matemáticas, Campus de Espinardo, Universidad de Murcia, 30100 Espinardo, Murcia, Spain.
Carolina Martínez-Riquelme
Affiliation:
Facultad de Matemáticas, Campus de Espinardo, Universidad de Murcia, 30100 Espinardo, Murcia, Spain. Email address: carolina.martinez7@um.es
José M. Ruiz
Affiliation:
Facultad de Matemáticas, Campus de Espinardo, Universidad de Murcia, 30100 Espinardo, Murcia, Spain. Email address: jmruizgo@um.es
Miguel A. Sordo
Affiliation:
Departamento Estadística e Investigación Operativa, Universidad de Cádiz, Facultad de Ciencias Económicas y Empresariales, 11002 Cádiz, Spain. Email address: mangel.sordo@uca.es
*
** Email address: belzunce@um.es
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this paper is twofold. On the one hand, we provide sufficient conditions for the excess wealth order. These conditions are based on properties of the quantile functions which are useful when the dispersive order does not hold. On the other hand, we study sufficient conditions for the comparison in the increasing convex order of spacings of generalized order statistics. These results will be combined to show how we can provide comparisons of quantities of interest in reliability and insurance.

Keywords

Excess wealth ordergeneralized order statisticsspacings
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 33 - 46
Copyright
Copyright © Applied Probability Trust 2016 

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

Asimit, A. V. and Jones, B. L. (2008). Asymptotic tail probabilities for large claims reinsurance of a portfolio of dependent risks. ASTIN Bull. 38, 147–159. CrossRefGoogle Scholar
Balakrishnan, N. and Rao, C. R. (eds) (1998a). Order Statistics: Theory and Methods (Handbook Statist. 16). North-Holland, Amsterdam. Google Scholar
Balakrishnan, N. and Rao, C. R. (eds) (1998b). Order Statistics: Applications (Handbook Statist. 17). North-Holland, Amsterdam. Google Scholar
Balakrishnan, N. and Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Prob. Eng. Inf. Sci. 27, 403–443. CrossRefGoogle Scholar
Balakrishnan, N., Belzunce, F., Sordo, M. A. and Suárez-Llorens, A. (2012). Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data. J. Multivariate Anal. 105, 45–54. 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, 103–110. CrossRefGoogle Scholar
Belzunce, F., Hu, T. and Khaledi, B.-E. (2003). Dispersion-type variability orders. Prob. Eng. Inf. Sci. 17, 305–334. CrossRefGoogle Scholar
Belzunce, F., Martínez-Riquelme, C. and Ruiz, J. M. (2014). A characterization and sufficient conditions for the total time on test transform order. TEST 23, 72–85. CrossRefGoogle Scholar
Belzunce, F., Mercader, J. A. and Ruiz, J. M. (2005). Stochastic comparisons of generalized order statistics. Prob. Eng. Inf. Sci. 19, 99–120. CrossRefGoogle Scholar
Belzunce, F., Ruiz, J. M. and Suárez-Llorens, A. (2008). On multivariate dispersion orderings based on the standard construction. Statist. Prob. Lett. 78, 271–281. CrossRefGoogle Scholar
Denuit, M. and Vermandele, C. (1999). Lorenz and excess wealth orders, with applications in reinsurance theory. Scand. Actuarial J. 1999, 170–185. CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin. CrossRefGoogle Scholar
Fernandez-Ponce, J. M., Kochar, S. C. and Muñoz-Perez, J. (1998). Partial orderings of distributions based on right-spread functions. J. Appl. Prob. 35, 221–228. CrossRefGoogle Scholar
Franco, M., Ruiz, J. M. and Ruiz, M. C. (2002). Stochastic orderings between spacings of generalized order statistics. Prob. Eng. Inf. Sci. 16, 471–484. CrossRefGoogle Scholar
Govindarajulu, Z. (1977). A class of distributions useful in life testing and reliability with applications to nonparametric testing. In The Theory and Applications of Reliability, Vol. I, Academic Press, New York, pp. 109–129. Google Scholar
Hankin, R. K. S. and Lee, A. (2006). A new family of non-negative distributions. Austral. N. Z. J. Statist. 48, 67–78. CrossRefGoogle Scholar
Hu, T. and Zhuang, W. (2005). Stochastic properties of p-spacings of generalized order statistics. Prob. Eng. Inf. Sci. 19, 257–276. CrossRefGoogle Scholar
Jiang, J. and Tang, Q. (2008). Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims. Insurance Math. Econom. 43, 431–436. CrossRefGoogle Scholar
Kamps, U. (1995a). A Concept of Generalized Order Statistics. Teubner, Stuttgart. CrossRefGoogle Scholar
Kamps, U. (1995b). A concept of generalized order statistics. J. Statist. Planning Infer. 48, 1–23. CrossRefGoogle Scholar
Kochar, S. C. (1990). Some partial ordering results on record values. Commun. Statist. Theory Meth. 19, 299–306. CrossRefGoogle Scholar
Kochar, S. and Xu, M. (2013). Excess wealth transform with applications. In Stochastic Orders in Reliability and Risk (Lecture Notes Statist. 208), Springer, New York, pp. 273–288. 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, 826–845. CrossRefGoogle Scholar
Kochar, S., Li, X. and Xu, M. (2007). Excess wealth order and sample spacings. Statist. Methodol. 4, 385–392. CrossRefGoogle Scholar
Li, X. (2005). A note on expected rent in auction theory. Operat. Res. Lett. 33, 531–534. CrossRefGoogle Scholar
Qiu, G. and Wang, J. (2007). Some comparisons between generalized order statistics. Appl. Math. J. Chinese Univ. 22, 325–333. CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1998). Two variability orders. Prob. Eng. Inf. Sci. 12, 1–23. CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York. CrossRefGoogle Scholar
Singpurwalla, N. D. and Gordon, A. S. (2014). Auditing Shaked and Shanthikumar's 'excess wealth'. Ann. Operat. Res. 212, 3–19. CrossRefGoogle Scholar
Sordo, M. A. (2008). Characterizations of classes of risk measures by dispersive orders. Insurance Math. Econom. 42, 1028–1034. CrossRefGoogle Scholar
Sordo, M. A. (2009). Comparing tail variability of risks by means of the excess wealth order. Insurance Math. Econom. 45, 466–469. CrossRefGoogle Scholar
Sordo, M. A., Suárez-Llorens, A. and Bello, A. J. (2015). Comparisons of conditional distributions in portfolios of dependent risks. Insurance Math. Econom. 61, 62–69. CrossRefGoogle Scholar
Thépaut, A. (1950). Une nouvelle forme de réassurance. Le traité d'excédent du coût moyen relatif (ECOMOR). Bull. Trimestriel Inst. Actuaires Français 49, 273–343. Google Scholar
Xie, H. and Hu, T. (2009). Ordering p-spacings of generalized order statistics revisited. Prob. Eng. Inf. Sci. 23, 1–16. CrossRefGoogle Scholar
Xie, H. and Hu, T. (2010). Some new results on multivariate dispersive ordering of generalized order statistics. J. Multivariate Anal. 101, 964–970. CrossRefGoogle Scholar
Xie, H. and Zhuang, W. (2011). Some new results on ordering of simple spacings of generalized order statistics. Prob. Eng. Inf. Sci. 25, 71–81. CrossRefGoogle Scholar
Zhuang, W. and Hu, T. (2009). Multivariate dispersive ordering of spacings of generalized order statistics. Appl. Math. Lett. 22, 968–974. CrossRefGoogle Scholar