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STOCHASTIC COMPARISONS OF GENERALIZED ORDER STATISTICS

Published online by Cambridge University Press:  01 January 2005

Félix Belzunce ,
José-Angel Mercader  and
José-María Ruiz
Félix Belzunce
Affiliation:
Departmento Estadística e Investigación Operativa, Universidad de Murcia, 30100 Espinardo (Murcia), Spain, E-mail: belzunce@um.es; mercader@um.es; jmruizgo@um.es
José-Angel Mercader
Affiliation:
Departmento Estadística e Investigación Operativa, Universidad de Murcia, 30100 Espinardo (Murcia), Spain, E-mail: belzunce@um.es; mercader@um.es; jmruizgo@um.es
José-María Ruiz
Affiliation:
Departmento Estadística e Investigación Operativa, Universidad de Murcia, 30100 Espinardo (Murcia), Spain, E-mail: belzunce@um.es; mercader@um.es; jmruizgo@um.es
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Abstract

In this article, we give several results on (multivariate and univariate) stochastic comparisons of generalized order statistics. We give conditions on the underlying distributions and the parameters on which the generalized order statistics are based, to obtain stochastic comparisons in the stochastic, dispersive, hazard rate, and likelihood ratio orders. Our results generalize some recent results for order statistics, record values, and generalized order statistics and provide some new results for other models such as k-record values and order statistics under multivariate imperfect repair.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 19 , Issue 1 , January 2005 , pp. 99 - 120
Copyright
© 2005 Cambridge University Press

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References

REFERENCES

Bartoszewicz, J. (1986). Dispersive ordering and the total time on test transform. Statistics and Probability Letters 4: 285–288.Google Scholar
Belzunce, F., Franco, M., Ruiz, J.M., & Ruiz, M.C. (2001). On partial orderings between coherent systems with different structures. Probability in the Engineering and Informational Sciences 15: 273–293.Google Scholar
Belzunce, F., Hu, T., & Khaledi, B.E. (2003). Dispersion-type variability orders. Probability in the Engineering and Informational Sciences 17: 305–334.Google Scholar
Belzunce, F., Lillo, R., Ruiz, J.M., & Shaked, M. (2001). Stochastic comparisons of nonhomogeneous processes. Probability in the Engineering and Informational Sciences 15: 199–224.Google Scholar
Belzunce, F. & Ruiz, J.M. (2002). Multivariate dispersive ordering of epoch times of nonhomogeneous Poisson processes. Journal of Applied Probability 39: 637–643.Google Scholar
Belzunce, F., Ruiz, J.M., & Ruiz, M.C. (2003). Multivariate properties of random vectors of order statistics. Journal of Statistical Planning and Inference 115: 413–424.Google Scholar
Belzunce, F. & Shaked, M. (2001). Stochastic comparisons of mixtures of convexly ordered distributions with applications in reliability theory. Statistics and Probability Letters 53: 363–372.Google Scholar
Belzunce, F. & Shaked, M. (2004). Failure profile of coherent systems, Naval Research Logistics 51: 477–490.Google Scholar
Boland, P.J., Hu, T., Shaked, M., & Shanthikumar, J.G. (2002). Stochastic ordering of order statistics II. In M. Dror, P. L'Ecuyer, & F. Szidarovszky (eds.), Modelling uncertainty: An examination of stochastic theory, methods and applications. Boston: Kluwer Academic, pp. 607–623.
Boland, P.J., Shaked, M., & Shanthikumar, J.G. (1998). Stochastic ordering of order statistics. In N. Balakrishnan & C.R. Rao (eds.), Handbook of statistics, vol. 16. Amsterdam: Elsevier Science, pp. 89–103.
Cramer, E. & Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58: 293–310.Google Scholar
Denuit, M., Lefèvre, C., & Shaked, M. (1998). The s-convex orders among real random variables, with applications. Mathematical Inequalities and Applications 1: 585–613.Google Scholar
Franco, M., Ruiz, J.M., & Ruiz, M.C. (2001). Stochastic orderings between spacings of generalized order statistics. Probability in the Engineering and Informational Sciences 16: 471–484.Google Scholar
Gupta, R.C. & Kirmani, S.N.U.A. (1988). Closure and monotonicity properties of nonhomogeneous Poisson processes and record values. Probability in the Engineering and Informational Sciences 2: 475–484.Google Scholar
Hu, T. (1996). Stochastic comparisons of order statistics under multivariate imperfect repair. Journal of Applied Probability 33: 156–163.Google Scholar
Hu, T. & Pan, X. (1999). Preservation of multivariate dependence under multivariate claim models. Insurance: Mathematics and Economics 25: 171–179.Google Scholar
Hu, T. & Wei, Y. (2001). Stochastic comparisons of spacings from restricted families of distributions. Statistics and Probability Letters 53: 91–99.Google Scholar
Kamps, U. (1994). Reliability properties of record values from non-identically distributed random variables. Communications in Statistics: Theory and Methods 23: 2101–2112.Google Scholar
Kamps, U. (1995). A concept of generalized order statistics. Journal of Statistical Planning and Inference 48: 1–23.Google Scholar
Kamps, U. (1995). A concept of generalized order statistics. Stuttgart: Taubner.
Khaledi, B.-E. & Kochar, S.C. (1999). Stochastic orderings between distributions and their sample spacings. Statistics and Probability Letters 44: 161–166.Google Scholar
Khaledi, B.-E. & Kochar, S.C. (2000). On dispersive ordering between order statistics in one-sample and two-sample problems. Statistics and Probability Letters 46: 257–261.Google Scholar
Kochar, S.C. (1999). On stochastic ordering between distributions and their sample spacings. Statistics and Probability Letters 42: 345–352.Google Scholar
Lewis, T. & Thompson, J.W. (1981). Dispersive distributions, and the connection between dispersivity and strong unimodality. Journal of Applied Probability 18: 76–90.Google Scholar
Lillo, R., Nanda, A.K., & Shaked, M. (2001). Preservation of some likelihood ratio orders by order statistics. Statistics and Probability Letters 51: 111–119.Google Scholar
Müller, A. & Stoyan, D. (2002). Comparisons methods for stochastic models and risks. West Sussex, UK: Wiley.
Nanda, A.K. & Shaked, M. (2002). The hazard rate and the reversed hazard rate orders, with applications to order statistics. Annals of the Institute of Statistical Mathematics 53: 853–864.Google Scholar
Pellerey, F., Shaked, M., & Zinn, J. (2000). Nonhomogeneous Poisson processes and logconcavity. Probability in the Engineering and Informational Sciences 14: 353–373.Google Scholar
Pfeifer, D. (1979). “Record values” in einem stochastischen Modell mit nicht-identischen Verteilungen. Dissertation, Aachen University of Technology.
Shaked, M. & Shanthikumar, J.G. (1986). Multivariate imperfect repair. Operations Research 34: 437–448.Google Scholar
Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. San Diego, CA: Academic Press.
Shaked, M. & Shanthikumar, J.G. (1998). Two variability orders. Probability in the Engineering and Informational Sciences 12: 1–23.Google Scholar
Shaked, M., Spizzichino, F., & Sutter, F. (2002). Nonhomogeneous birth processes and I∞-spherical densities, with applications in reliability theory. Probability in the Engineering and Informational Sciences 16: 271–288.Google Scholar
Shaked, M. & Szekli, R. (1995). Comparison of replacement policies via point processes. Advances in Applied Probability 27: 1079–1103.Google Scholar