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Stochastic Orders for Spacings of Heterogeneous Exponential Random Variables

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  • Kochar, Subhash C
  • Korwar, Ramesh

Abstract

We obtain some new results on normalized spacings of independent exponential random variables with possibly different scale parameters. It is shown that the density functions of the individual normalized spacings in this case are mixtures of exponential distributions and, as a result, they are log-convex (and, hence, DFR). G. Pledger and F. Proschan (Optimizing Methods in Statistics(J. S. Rustagi, Ed.), pp. 89-113, Academic Press, New York, 1971), have shown, with the help of a counterexample, that in a sample of size 3 the survival function of the last spacing is not Schur convex. We show that, however, this is true for the second spacing for all sample sizes. G. Pledger and F. Proschan (ibid.) also prove that the spacings are stochastically larger when the scale parameters are unequal than when they are all equal. We strengthen this result from stochastic ordering to likelihood ratio ordering. Some new results on dispersive ordering between the normalized spacings have also been obtained.

Suggested Citation

  • Kochar, Subhash C & Korwar, Ramesh, 1996. "Stochastic Orders for Spacings of Heterogeneous Exponential Random Variables," Journal of Multivariate Analysis, Elsevier, vol. 57(1), pages 69-83, April.
    • Handle: RePEc:eee:jmvana:v:57:y:1996:i:1:p:69-83
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    Cited by:

    1. Huaihou Chen & Taizhong Hu, 2008. "Multivariate likelihood ratio orderings between spacings of heterogeneous exponential random variables," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 68(1), pages 17-29, June.
    2. Paltanea, Eugen, 2011. "Bounds for mixtures of order statistics from exponentials and applications," Journal of Multivariate Analysis, Elsevier, vol. 102(5), pages 896-907, May.
    3. Fang, Rui & Ding, Weiyong, 2018. "On relative log-concavity and stochastic comparisons," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 91-98.
    4. Dolati, Ali & Genest, Christian & Kochar, Subhash C., 2008. "On the dependence between the extreme order statistics in the proportional hazards model," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 777-786, May.
    5. Kochar, Subhash & Ma, Chunsheng, 1999. "Dispersive ordering of convolutions of exponential random variables," Statistics & Probability Letters, Elsevier, vol. 43(3), pages 321-324, July.
    6. Fischer, T. & Balakrishnan, N. & Cramer, E., 2008. "Mixture representation for order statistics from INID progressive censoring and its applications," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 1999-2015, October.
    7. Zhang, Zhengcheng & Yang, Luyi & Yang, Yonghong, 2022. "The conditional spacings and their stochastic properties," Statistics & Probability Letters, Elsevier, vol. 186(C).
    8. Wen, Songqiao & Lu, Qingshu & Hu, Taizhong, 2007. "Likelihood ratio orderings of spacings of heterogeneous exponential random variables," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 743-756, April.
    9. Peng Zhao & N. Balakrishnan, 2011. "Dispersive ordering of fail-safe systems with heterogeneous exponential components," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(2), pages 203-210, September.
    10. Zhao, Peng & Li, Xiaohu & Balakrishnan, N., 2009. "Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 952-962, May.
    11. Torrado Robles, Nuria & Lillo Rodríguez, Rosa Elvira & Wiper, Michael Peter, 2009. "On the Conjecture of Kochar and Korwar," DES - Working Papers. Statistics and Econometrics. WS ws092108, Universidad Carlos III de Madrid. Departamento de Estadística.
    12. Torrado, Nuria & Lillo, Rosa E. & Wiper, Michael P., 2010. "On the conjecture of Kochar and Korwar," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1274-1283, May.
    13. N. Balakrishnan & Erhard Cramer, 2008. "Progressive censoring from heterogeneous distributions with applications to robustness," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(1), pages 151-171, March.
    14. Jongwoo Jeon & Subhash Kochar & Chul Park, 2006. "Dispersive ordering—Some applications and examples," Statistical Papers, Springer, vol. 47(2), pages 227-247, March.
    15. Lihong, Sun & Xinsheng, Zhang, 2005. "Stochastic comparisons of order statistics from gamma distributions," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 112-121, March.
    16. Genest, Christian & Kochar, Subhash C. & Xu, Maochao, 2009. "On the range of heterogeneous samples," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1587-1592, September.
    17. Chu, Yu-Ming & Xia, Wei-Feng & Zhang, Xiao-Hui, 2012. "The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 412-421.
    18. Torrado, Nuria & Lillo, Rosa E., 2013. "Likelihood ratio order of spacings from two heterogeneous samples," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 338-348.
    19. Arendarczyk, Marek & Kozubowski, Tomasz. J. & Panorska, Anna K., 2018. "The joint distribution of the sum and the maximum of heterogeneous exponential random variables," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 10-19.
    20. Li, Chen & Li, Xiaohu, 2015. "Likelihood ratio order of sample minimum from heterogeneous Weibull random variables," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 46-53.
    21. Subhash Kochar & Ramesh Korwar, 2001. "On Random Sampling Without Replacement from a Finite Population," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(3), pages 631-646, September.
    22. Ding, Weiyong & Da, Gaofeng & Zhao, Peng, 2013. "On sample ranges from two sets of heterogenous random variables," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 63-73.
    23. Xu, Maochao & Balakrishnan, N., 2012. "On the sample ranges from heterogeneous exponential variables," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 1-9.
    24. Torrado Robles, Nuria & Lillo Rodríguez, Rosa Elvira, 2012. "Comparisons among spacings from two populations," DES - Working Papers. Statistics and Econometrics. WS 13958, Universidad Carlos III de Madrid. Departamento de Estadística.

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