Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-20T04:07:07.567Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME NEW RESULTS ON THE LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER GAMMA MODELS

Published online by Cambridge University Press:  16 July 2015

Peng Zhao ,
Yanni Hu  and
Yiying Zhang
Peng Zhao
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China E-mail: zhaop@jsnu.edu.cn
Yanni Hu
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Yiying Zhang
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we carry out stochastic comparisons of the largest order statistics arising from multiple-outlier gamma models with different both shape and scale parameters in the sense of various stochastic orderings including the likelihood ratio order, star order and dispersive order. It is proved, among others, that the weak majorization order between the scale parameter vectors along with the majorization order between the shape parameter vectors imply the likelihood ratio order between the largest order statistics. A quite general sufficient condition for the star order is presented. The new results established here strengthen and generalize some of the results known in the literature. Numerical examples and applications are also provided to explicate the theoretical results.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 29 , Issue 4 , October 2015 , pp. 597 - 621
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Ahmed, A.N., Alzaid, A., Bartoszewicz, J., & Kochar, S.C. (1986). Dispersive and superadditive ordering. Advances in Applied Probability 18: 1019–1022.Google Scholar
2. Balakrishnan, N. (2007). Permanents, order statistics, outliers, and robustness. Revista Matemática Complutense 20: 7–107.Google Scholar
3. Balakrishnan, N. & Rao, C.R. (1998a). Handbook of Statistics. Vol. 16: Order Statistics: Theory and Methods. Amsterdam: Elsevier.Google Scholar
4. Balakrishnan, N. & Rao, C.R. (1998b). Handbook of Statistics. Vol. 17: Order Statistics: Applications. Amsterdam: Elsevier.Google Scholar
5. Barlow, R.E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Maryland: Silver Spring Google Scholar
6. Boland, P.J., EL-Neweihi, E., & Proschan, F. (1994). Applications of the hazard rate ordering in reliability and order statistics. Journal of Applied Probability 31: 180–192.Google Scholar
7. Bon, J.L. & Pǎltǎnea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5: 185–192.Google Scholar
8. Bon, J.L. & Pǎltǎnea, E. (2006). Comparisons of order statistics in a random sequence to the same statistics with i.i.d. variables. ESAIM: Probability and Statistics 10: 1–10.Google Scholar
9. David, H.A. & Nagaraja, H.N. (2003). Order Statistics. 3rd edn., Hoboken, NJ: John Wiley & Sons.Google Scholar
10. Dykstra, R., Kochar, S.C., & Rojo, J. (1997). Stochastic comparisons of parallel systems of heterogeneous exponential components. Journal of Statistical Planning and Inference 65: 203–211.Google Scholar
11. Hu, T. (1995). Monotone coupling and stochastic ordering of order statistics. System Science and Mathematical Science (English Series). 8: 209–214.Google Scholar
12. Joo, S. & Mi, J. (2010). Some properties of hazard rate functions of systems with two components. Journal of Statistical Planning and Inference 140: 444–453.Google Scholar
13. Khaledi, B.-E. & Kochar, S.C. (2000). Some new results on stochastic comparisons of parallel systems. Journal of Applied Probability 37: 283–291.Google Scholar
14. Khaledi, B.-E., Farsinezhad, S., & Kochar, S.C. (2011). Stochastic comparisons of order statistics in the scale models. Journal of Statistical Planning and Inference 141: 276–286.Google Scholar
15. Kochar, S.C. & Rojo, J. (1996). Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions. Journal of Multivariate Analysis 59: 272–281.Google Scholar
16. Kochar, S.C. & Xu, M. (2007). Stochastic comparisons of parallel systems when components have proportional hazard rates. Probability in the Engineering and Informational Science 21: 597–609.Google Scholar
17. Kochar, S.C. & Xu, M. (2011). On the skewness of order statistics in the multiple-outlier models. Journal of Applied Probability 48: 271–284.Google Scholar
18. Mao, T. & Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Probability in the Engineering and Informational Science 24: 245–262.Google Scholar
19. Marshall, A.W. & Olkin, I. (2007). Life Distributions. New York: Springer.Google Scholar
20. Marshall, A.W., Olkin, I., & Arnold, B.C. (2011). Inequalities: Theory of Majorization and Its Applications: 2nd edn., New York: Springer-Verlag.Google Scholar
21. Misra, N. & van der Meulen, E.C. (2003) On stochastic properties of m-spacings. Journal of Statistical Planning and Inferences 115: 683–697.Google Scholar
22. Pǎltǎnea, E. (2008). On the comparison in hazard rate ordering of fail-safe systems. Journal of Statistical Planning and Inference 138: 1993–1997.Google Scholar
23. Pledger, P. & Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Rustagi, J.S. (ed.), Optimizing Methods in Statistics, New York: Academic Press, pp. 89–113.Google Scholar
24. Proschan, F. & Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. Journal of Multivariate Analysis 6: 608–616.Google Scholar
25. Saunders, I. W. & Moran, P. A. (1978). On the quantiles of the gamma and F distributions. Journal of Applied Probability 15: 426–432.Google Scholar
26. Shaked, M. & Shanthikumar, J.G. (1992). Optimal allocation of resources to nodes of parallel and series systems. Advances in Applied Probability 24: 894–914.Google Scholar
27. Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. New York: Springer-Verlag.Google Scholar
28. Zhao, P. (2011). On parallel systems with heterogeneous gamma components. Probability in the Engineering and Informational Science 25: 369–391.Google Scholar
29. Zhao, P. & Balakrishnan, N. (2011a). Some characterization results for parallel systems with two heterogeneous exponential components. Statistics 45: 593–604.Google Scholar
30. Zhao, P. & Balakrishnan, N. (2011b). New results on comparisons of parallel systems with heterogeneous gamma components. Statistics and Probability Letters 81: 36–44.Google Scholar
31. Zhao, P. & Balakrishnan, N. (2014). Comparisons of largest order statistics from multiple outlier gamma models. Methodology and Computing in Applied Probability DOI:10.1007/s11009-013-9377-0.Google Scholar
32. Zhao, P. & Zhang, Y. (2014). On the maxima of heterogeneous gamma variables with different shape and scale parameters. Metrika 77: 811–836.Google Scholar