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Stochastic Comparisons of Spacings from Heterogeneous Samples

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Advances in Directional and Linear Statistics

Abstract

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.

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References

  1. Avérous J, Genest C, Kochar SC (2005) On the dependence structure of order statistics. J Multivar Anal 94:159–171

    Article  MATH  Google Scholar 

  2. Balakrishnan N, Rao CR (1998) Handbook of statistics 16-order statistics: Theory and methods. Elsevier, New York

    Google Scholar 

  3. Barlow RE, Proschan F (1966) Inequalities for linear combinations of order statistics from restricted families. Ann Math Statist 37:1574–1592

    Article  MATH  MathSciNet  Google Scholar 

  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 10:1–10

    Article  MATH  MathSciNet  Google Scholar 

  5. Burkschat M (2009) Multivariate dependence of spacings of generalized order statistics. J Multivar Anal 100:1093–1106

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen H, Hu T (2008) Multivariate likelihood ratio orderings between spacings of heterogeneous exponential random variables. Metrika 68:17–29

    Article  MathSciNet  Google Scholar 

  7. Dolati A, Genest C, Kochar SC (2008) On the dependence between the extreme order statistics in the proportional hazards model. J Multivar Anal 99:777–786

    Article  MATH  MathSciNet  Google Scholar 

  8. Epstein B (1960) Estimation from life test data. Technometric 2:447–454

    Article  MATH  Google Scholar 

  9. Fernández-Ponce JM, Kochar SC, Muñoz-Perez J (1998) Partial orderings of distributions based on right-spread functions. J Appl Probab 35:221–228

    Article  MATH  MathSciNet  Google Scholar 

  10. Genest C, Kochar S, Xu M (2009) On the range of heterogeneous samples. J Multivar Anal 100:1587–1592

    Article  MATH  MathSciNet  Google Scholar 

  11. Hollander M, Proschan F, Sethuraman J (1977) Functions decreasing in transposition and their applications in ranking problems. Ann Stat 4:722–733

    Article  MathSciNet  Google Scholar 

  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 37:2506–2515

    Article  MATH  MathSciNet  Google Scholar 

  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 35:979–988

    Article  MATH  MathSciNet  Google Scholar 

  14. Jammalamadaka SR, Goria MN (2004) A test of goodness of git based on Gini’s index of spacings. Stat Prob Lett 68:177–187

    Article  MATH  MathSciNet  Google Scholar 

  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 15:719–729

    Article  MATH  MathSciNet  Google Scholar 

  16. Khaledi B, Kochar SC (2000a) Dependence among spacings. Probab Eng Inform Sci 14: 461–472

    Article  MATH  MathSciNet  Google Scholar 

  17. Khaledi B, Kochar SC (2000b) Sample range-some stochastic comparisons results. Calcutta Stat Assoc Bull 50:283–291

    MATH  MathSciNet  Google Scholar 

  18. Khaledi B, Kochar SC (2001) Stochastic properties of spacings in a single-outlier exponential model. Probab Eng Inform Sci 15:401–408

    Article  MATH  MathSciNet  Google Scholar 

  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 201–216

    Google Scholar 

  20. Kochar SC, Kirmani S (1995) Some results on normalized spacings from restricted families of distributions. 46:47–57

    MATH  MathSciNet  Google Scholar 

  21. Kochar SC, Korwar R (1996) Stochastic orders for spacings of heterogeneous exponential random variables. J Multivar Anal 59:272–281

    Article  MATH  MathSciNet  Google Scholar 

  22. Kochar SC, Rojo J (1996) Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions. J Multivar Anal 57:69–83

    Article  MATH  MathSciNet  Google Scholar 

  23. Kochar SC, Xu M (2007) Stochastic comparisons of parallel systems when components have proportional hazard rates. Probab Eng Inform Sci 21:597–609

    MATH  MathSciNet  Google Scholar 

  24. Marshall AW, Olkin I (2007) Life distributions. Springer, New York

    MATH  Google Scholar 

  25. Misra N, van der Meulen EC (2003) On stochastic properties of m-spacings. J Stat Plan Inference 115:683–697

    Article  MATH  Google Scholar 

  26. Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, New York

    MATH  Google Scholar 

  27. Nappo G, Spizzichino F (1998) Ordering properties of the TTT-plot of lifetimes with Schur joint densities. Stat Prob Lett 39:195–203

    Article  MATH  MathSciNet  Google Scholar 

  28. Pǎltǎnea E (2008) On the comparison in hazard rate ordering of fail-safe systems. J Stat Plan Inference 138:1993–1997

    Article  Google Scholar 

  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 89–113

    Google Scholar 

  30. Proschan F, Sethuraman J (1976) Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. J Multivariate Anal 6:608–616

    Article  MATH  MathSciNet  Google Scholar 

  31. Robertson T, Wright FT (1982) On measuring the conformity of a parameter set to a trend, with applications. Ann Stat 4:1234–1245

    Article  MathSciNet  Google Scholar 

  32. Shaked M, Shantikumar JG (1998) Two variability orders. Probab Eng Inform Sci 12:1–23

    Article  MATH  Google Scholar 

  33. Shaked M, Shanthikumar JG (2007) Stochastic orders and their applications. Springer, New York

    Google Scholar 

  34. Shanthikumar JG, Yao DD (1991) Bivariate characterization of some stochastic order relations. Adv Appl Prob 23:642–659

    Article  MATH  MathSciNet  Google Scholar 

  35. Wen S, Lu Q, Hu T (2007) Likelihood ratio orderings of spacings of heterogeneous exponential random variables. J Multivar Anal 98:743–756

    Article  MATH  MathSciNet  Google Scholar 

  36. Xu M, Li X, Zhao P, Li Z (2007) Likelihood ratio order of m-spacings in multipleoutlier models. Commun Stat Theory Methods 36:1507–1525

    Article  MATH  MathSciNet  Google Scholar 

  37. Xu M, Li X (2008) Some further results on the winner’s rent in the second-price business auction. Sankhya 70:124–133

    MATH  MathSciNet  Google Scholar 

  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.2009.02.006

    Google Scholar 

  39. Zhao P, Li X (2009) Stochastic order of sample range from heterogeneous exponential random variables. Probab Eng Inform Sci 23:17–29

    Article  MATH  Google Scholar 

  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 100:952–962

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Portland State University, Portland, OR, 97201, USA

    Subhash Kochar

Authors
  1. Subhash Kochar
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  2. Maochao Xu
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Corresponding author

Correspondence to Subhash Kochar .

Editor information

Editors and Affiliations

  1. , Statistical Science, Cornell University, Comstock Hall 1190, Ithaca, NY 14853, 14853, USA

    Martin T. Wells

  2. , Applied Statistics Unit, Indian Statistical Institute, Barrackpore Trunk Road 203, Kolkata, 700035, India

    Ashis SenGupta

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Kochar, S., Xu, M. (2011). Stochastic Comparisons of Spacings from Heterogeneous Samples. In: Wells, M., SenGupta, A. (eds) Advances in Directional and Linear Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2628-9_8

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