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The total time on test transform and the excess wealth stochastic orders of distributions

Part of: Survival analysis and censored data Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Subhash C. Kochar ,
Xiaohu Li  and
Moshe Shaked
Subhash C. Kochar*
Affiliation:
Indian Statistical Institute
Xiaohu Li*
Affiliation:
Lanzhou University
Moshe Shaked*
Affiliation:
University of Arizona
*
∗ Postal address: Stat-Math Unit, Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India.
∗∗ Postal address: Department of Mathematics, Lanzhou University, Lanzhou 730000, People's Republic of China.
∗∗∗ Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA. Email address: shaked@math.arizona.edu
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Abstract

For nonnegative random variables X and Y we write X ≤TTTY if ∫0F-1(p)(1-F(x))dx ≤ ∫0G-1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

Keywords

Excess wealth orderright-spread orderLorenz orderNBUEincreasing convex and concave ordersseries and parallel systemsHNBUEeconomic inequality measureempirical TTT transformtest for ‘more NBUE’

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 62N05: Reliability and life testing
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 826 - 845
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

Supported by NSFC under grants TY 10126014 and 10201010.

References

Barlow, R. E. (1979). Geometry of the total time on test transform. Naval Res. Logistics Quart. 26, 393–402.Google Scholar
Barlow, R. E. and Campo, R. (1975). Total time on test processes and applications to failure data analysis. In Reliability and Fault Tree Analysis, eds Barlow, R. E., Fussell, J. B. and Singpurwalla, N. D., SIAM, Philadelphia, pp. 451–481.Google Scholar
Barlow, R. E. and Doksum, K. A. (1972). Isotonic tests for convex orderings. In Proc. 6th Berkeley Symp. Math. Statist. Prob., Vol. I, eds Le Cam, L. M., Neyman, J. and Scott, E. L., University of California Press, Berkeley, pp. 293–323.Google Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference Under Order Restrictions. John Wiley, New York.Google Scholar
Bartoszewicz, J. (1985). Moment inequalities for order statistics from ordered families of distributions. Metrika 32, 383–389.Google Scholar
Bartoszewicz, J. (1986). Dispersive ordering and the total time on test transformation. Statist. Prob. Lett. 4, 285–288.Google Scholar
Bartoszewicz, J. (1995). Stochastic order relations and the total time on test transform. Statist. Prob. Lett. 22, 103–110.Google Scholar
Bartoszewicz, J. (1998). Applications of a general composition theorem to the star order of distributions. Statist. Prob. Lett. 38, 1–9.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
Bergman, B. (1979). On age replacement and the total time on test concept. Scand. J. Statist. 6, 161–168.Google Scholar
Bhattacharjee, M. C. and Krishnaji, N. (1984). DFR and other heavy tail properties in modelling the distribution of land and some alternative measures of inequality. In Statistics: Applications and New Directions, eds Ghosh, J. K. and Roy, J., Indian Statistical Institute, Calcutta, pp. 100–115.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-Perez, J. (1998). Partial orderings of distributions based on right-spread functions. J. Appl. Prob. 35, 221–228.Google Scholar
Gerlach, B. (1988). A new test for whether (F) is ‘more NBU’ than (G). Metrika 17, 79–86.Google Scholar
Hollander, M. and Proschan, F. (1975). Tests for mean residual life. Biometrika 62, 585–593.Google Scholar
Høyland, A. and Rausand, M. (1994). System Reliability Theory: Models and Statistical Methods. John Wiley, New York.Google Scholar
Hürlimann, W., (2002). Analytical evaluation of economic risk capital for portfolios of gamma risks. A unified approach to conditional value-at-risk, with application to RAROC and RARAROC. Tech. Rep., Value and Risk Management, Winterthur Life and Pensions, Winterthur, Switzerland.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
Klefsjö, B., (1983). Testing exponentiality against HNBUE. Scand. J. Statist. 10, 65–75.Google Scholar
Klefsjö, B., (1984). Reliability interpretations of some concepts from economics. Naval Res. Logistics Quart. 31, 301–308.Google Scholar
Klefsjö, B., (1991). TTT-plotting—a tool for both theoretical and practical problems. J. Statist. Planning Infer. 29, 111–124.Google Scholar
Kochar, S. C. (1989). On extensions of DMRL and related partial orderings of life distributions. Commun. Statist. Stoch. Models 5, 235–245.Google Scholar
Kochar, S. C. and Carrière, K. C. (1997). Connections among various variability orderings. Statist. Prob. Lett. 35, 327–333.Google Scholar
Kochar, S. C. and Wiens, D. P. (1987). Partial orderings of life distributions with respect to their aging properties. Naval Res. Logistics 34, 823–829.Google Scholar
Li, X., Li, Z. and Jing, B.-Y. (2000). Some results about NBUC class of life distributions. Statist. Prob. Lett. 46, 229–237.Google Scholar
Rojo, J. and He, G. Z. (1991). New properties and characterizations of the dispersive ordering. Statist. Prob. Lett. 11, 365–372.Google Scholar
Ross, S. M. (1996). Stochastic Processes, 2nd edn. John Wiley. New York.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, San Diego, CA.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1998). Two variability orders. Prob. Eng. Inf. Sci. 12, 1–23.Google Scholar