Stochastic Comparisons of Spacings from Heterogeneous Samples



In this paper we review some of the recently obtained results in the area of stochastic comparisons of sample spacings when the observations are not necessarily identically distributed. A few new results on necessary and sufficient conditions for various stochastic orderings among spacings are also given. The paper is concluded with some examples and applications.


Hazard Rate Stochastic Order Exponential Random Variable Commun Stat Theory Method Multivar Anal 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avérous J, Genest C, Kochar SC (2005) On the dependence structure of order statistics. J Multivar Anal 94:159–171MATHCrossRefGoogle Scholar
  2. 2.
    Balakrishnan N, Rao CR (1998) Handbook of statistics 16-order statistics: Theory and methods. Elsevier, New YorkGoogle Scholar
  3. 3.
    Barlow RE, Proschan F (1966) Inequalities for linear combinations of order statistics from restricted families. Ann Math Statist 37:1574–1592MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bon JL, Pǎltǎnea E (2006) Comparison of order statistics in a random sequence to the same statistics with i.i.d. variables. ESAIM Probab Stat 10:1–10MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Burkschat M (2009) Multivariate dependence of spacings of generalized order statistics. J Multivar Anal 100:1093–1106MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen H, Hu T (2008) Multivariate likelihood ratio orderings between spacings of heterogeneous exponential random variables. Metrika 68:17–29CrossRefMathSciNetGoogle Scholar
  7. 7.
    Dolati A, Genest C, Kochar SC (2008) On the dependence between the extreme order statistics in the proportional hazards model. J Multivar Anal 99:777–786MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Epstein B (1960) Estimation from life test data. Technometric 2:447–454MATHCrossRefGoogle Scholar
  9. 9.
    Fernández-Ponce JM, Kochar SC, Muñoz-Perez J (1998) Partial orderings of distributions based on right-spread functions. J Appl Probab 35:221–228MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Genest C, Kochar S, Xu M (2009) On the range of heterogeneous samples. J Multivar Anal 100:1587–1592MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hollander M, Proschan F, Sethuraman J (1977) Functions decreasing in transposition and their applications in ranking problems. Ann Stat 4:722–733CrossRefMathSciNetGoogle Scholar
  12. 12.
    Hu T, Lu Q, Wen S (2008) Some new results on likelihood ratio orderings for spacings of heterogeneous exponential random variables. Commun Stat Theory Methods 37:2506–2515MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hu T, Wang F, Zhu Z (2006) Stochastic comparisons and dependence of spacings from two samples of exponential random variables. Commun Stat Theory Methods 35:979–988MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jammalamadaka SR, Goria MN (2004) A test of goodness of git based on Gini’s index of spacings. Stat Prob Lett 68:177–187MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Jammalamadaka SR, Taufer E (2003) Testing exponentiality by comparing the empirical distribution function of the normalized spacings with that of the original data. J Nonparametr Stat 15:719–729MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Khaledi B, Kochar SC (2000a) Dependence among spacings. Probab Eng Inform Sci 14: 461–472MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Khaledi B, Kochar SC (2000b) Sample range-some stochastic comparisons results. Calcutta Stat Assoc Bull 50:283–291MATHMathSciNetGoogle Scholar
  18. 18.
    Khaledi B, Kochar SC (2001) Stochastic properties of spacings in a single-outlier exponential model. Probab Eng Inform Sci 15:401–408MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kochar SC (1998) Stochastic comparisons of spacings and order statistics. In: Basu AP, Basu SK, Mukhopadhyay S (eds) Frontiers in reliability. World Scientific, Singapore, pp 201–216Google Scholar
  20. 20.
    Kochar SC, Kirmani S (1995) Some results on normalized spacings from restricted families of distributions. 46:47–57MATHMathSciNetGoogle Scholar
  21. 21.
    Kochar SC, Korwar R (1996) Stochastic orders for spacings of heterogeneous exponential random variables. J Multivar Anal 59:272–281MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kochar SC, Rojo J (1996) Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions. J Multivar Anal 57:69–83MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kochar SC, Xu M (2007) Stochastic comparisons of parallel systems when components have proportional hazard rates. Probab Eng Inform Sci 21:597–609MATHMathSciNetGoogle Scholar
  24. 24.
    Marshall AW, Olkin I (2007) Life distributions. Springer, New YorkMATHGoogle Scholar
  25. 25.
    Misra N, van der Meulen EC (2003) On stochastic properties of m-spacings. J Stat Plan Inference 115:683–697MATHCrossRefGoogle Scholar
  26. 26.
    Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, New YorkMATHGoogle Scholar
  27. 27.
    Nappo G, Spizzichino F (1998) Ordering properties of the TTT-plot of lifetimes with Schur joint densities. Stat Prob Lett 39:195–203MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Pǎltǎnea E (2008) On the comparison in hazard rate ordering of fail-safe systems. J Stat Plan Inference 138:1993–1997CrossRefGoogle Scholar
  29. 29.
    Pledger P, Proschan F (1971) Comparisons of order statistics and of spacings from heterogeneous distributions. In: Rustagi JS (ed) Optimizing methods in statistics. Academic, New York, pp 89–113Google Scholar
  30. 30.
    Proschan F, Sethuraman J (1976) Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J Multivariate Anal 6:608–616MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Robertson T, Wright FT (1982) On measuring the conformity of a parameter set to a trend, with applications. Ann Stat 4:1234–1245CrossRefMathSciNetGoogle Scholar
  32. 32.
    Shaked M, Shantikumar JG (1998) Two variability orders. Probab Eng Inform Sci 12:1–23MATHCrossRefGoogle Scholar
  33. 33.
    Shaked M, Shanthikumar JG (2007) Stochastic orders and their applications. Springer, New YorkGoogle Scholar
  34. 34.
    Shanthikumar JG, Yao DD (1991) Bivariate characterization of some stochastic order relations. Adv Appl Prob 23:642–659MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Wen S, Lu Q, Hu T (2007) Likelihood ratio orderings of spacings of heterogeneous exponential random variables. J Multivar Anal 98:743–756MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Xu M, Li X, Zhao P, Li Z (2007) Likelihood ratio order of m-spacings in multipleoutlier models. Commun Stat Theory Methods 36:1507–1525MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Xu M, Li X (2008) Some further results on the winner’s rent in the second-price business auction. Sankhya 70:124–133MATHMathSciNetGoogle Scholar
  38. 38.
    Zhao P, Balakrishnan N (2009) Characterization of MRL order of fail-safe systems with heterogeneous exponential components. J Stat Plan Inference, doi:10.1016/j.jspi.2009.02.006Google Scholar
  39. 39.
    Zhao P, Li X (2009) Stochastic order of sample range from heterogeneous exponential random variables. Probab Eng Inform Sci 23:17–29MATHCrossRefGoogle Scholar
  40. 40.
    Zhao P, Li X, Balakrishnan N (2009) Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables. J Multivar Anal 100:952–962MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsPortland State UniversityPortlandUSA

Personalised recommendations