Skip to main content Accessibility help
×
Home

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

  • Peng Zhao (a1), Yanni Hu (a2) and Yiying Zhang (a2)

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.

Copyright

References

Hide All
1. Ahmed, A.N., Alzaid, A., Bartoszewicz, J., & Kochar, S.C. (1986). Dispersive and superadditive ordering. Advances in Applied Probability 18: 10191022.
2. Balakrishnan, N. (2007). Permanents, order statistics, outliers, and robustness. Revista Matemática Complutense 20: 7107.
3. Balakrishnan, N. & Rao, C.R. (1998a). Handbook of Statistics. Vol. 16: Order Statistics: Theory and Methods. Amsterdam: Elsevier.
4. Balakrishnan, N. & Rao, C.R. (1998b). Handbook of Statistics. Vol. 17: Order Statistics: Applications. Amsterdam: Elsevier.
5. Barlow, R.E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Maryland: Silver Spring
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: 180192.
7. Bon, J.L. & Pǎltǎnea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5: 185192.
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: 110.
9. David, H.A. & Nagaraja, H.N. (2003). Order Statistics. 3rd edn., Hoboken, NJ: John Wiley & Sons.
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: 203211.
11. Hu, T. (1995). Monotone coupling and stochastic ordering of order statistics. System Science and Mathematical Science (English Series). 8: 209214.
12. Joo, S. & Mi, J. (2010). Some properties of hazard rate functions of systems with two components. Journal of Statistical Planning and Inference 140: 444453.
13. Khaledi, B.-E. & Kochar, S.C. (2000). Some new results on stochastic comparisons of parallel systems. Journal of Applied Probability 37: 283291.
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: 276286.
15. Kochar, S.C. & Rojo, J. (1996). Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions. Journal of Multivariate Analysis 59: 272281.
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: 597609.
17. Kochar, S.C. & Xu, M. (2011). On the skewness of order statistics in the multiple-outlier models. Journal of Applied Probability 48: 271284.
18. Mao, T. & Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Probability in the Engineering and Informational Science 24: 245262.
19. Marshall, A.W. & Olkin, I. (2007). Life Distributions. New York: Springer.
20. Marshall, A.W., Olkin, I., & Arnold, B.C. (2011). Inequalities: Theory of Majorization and Its Applications: 2nd edn., New York: Springer-Verlag.
21. Misra, N. & van der Meulen, E.C. (2003) On stochastic properties of m-spacings. Journal of Statistical Planning and Inferences 115: 683697.
22. Pǎltǎnea, E. (2008). On the comparison in hazard rate ordering of fail-safe systems. Journal of Statistical Planning and Inference 138: 19931997.
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. 89113.
24. Proschan, F. & Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. Journal of Multivariate Analysis 6: 608616.
25. Saunders, I. W. & Moran, P. A. (1978). On the quantiles of the gamma and F distributions. Journal of Applied Probability 15: 426432.
26. Shaked, M. & Shanthikumar, J.G. (1992). Optimal allocation of resources to nodes of parallel and series systems. Advances in Applied Probability 24: 894914.
27. Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. New York: Springer-Verlag.
28. Zhao, P. (2011). On parallel systems with heterogeneous gamma components. Probability in the Engineering and Informational Science 25: 369391.
29. Zhao, P. & Balakrishnan, N. (2011a). Some characterization results for parallel systems with two heterogeneous exponential components. Statistics 45: 593604.
30. Zhao, P. & Balakrishnan, N. (2011b). New results on comparisons of parallel systems with heterogeneous gamma components. Statistics and Probability Letters 81: 3644.
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.
32. Zhao, P. & Zhang, Y. (2014). On the maxima of heterogeneous gamma variables with different shape and scale parameters. Metrika 77: 811836.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed