Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T13:03:15.070Z Has data issue: false hasContentIssue false
  • English
  • Français

REVISITING MULTIVARIATE LIKELIHOOD RATIO ORDERING RESULTS FOR ORDER STATISTICS

Published online by Cambridge University Press:  17 May 2011

Félix Belzunce ,
Selma Gurler  and
José M. Ruiz
Félix Belzunce
Affiliation:
Departmento de Estadística e Investigación Operativa, Universidad de Murcia, Facultad de Matemáticas, Campus de Espinardo, 30100 Espinardo (Murcia), Spain E-mail: belzunce@um.es
Selma Gurler
Affiliation:
Department of Statistics, Dokuz Eylul University, Faculty of Science, Tinaztepe Campus, 35160 Buca, Izmir, Turkey E-mail: selma.erdogan@deu.edu.tr
José M. Ruiz
Affiliation:
Departmento de Estadística e Investigación Operativa, Universidad de Murcia, Facultad de Matemáticas, Campus de Espinardo, 30100 Espinardo (Murcia), Spain E-mail: jmruizgo@um.es
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we establish some results concerning the likelihood ratio order of random vectors of order statistics in the case of independent but not necessarily identically distributed observations and for the case of possible dependent observations. Applications of these results to provide comparisons of conditional order statistics are also given.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 25 , Issue 3 , July 2011 , pp. 355 - 368
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

1.Asadi, M. (2006). On the mean past lifetime of the components of a parallel system. Journal of Statistical Planning and Inference 136: 1197–1206.Google Scholar
2.Asadi, M. & Bairamov, I. (2005). A note on the mean residual life function of a parallel system. Communications in Statistics – Theory and Methods 34: 475–484.CrossRefGoogle Scholar
3.Balakrishnan, N. (2007). Permanents, order statistics, outliers and robustness. Revista Matemática Complutense 20: 7–107.Google Scholar
4.Balakrishnan, N., Belzunce, F., Hami, N. & Khaledi, B.E. (2010). Univariate and multivariate like-lihood ratio ordering of generalized order statistics and associated conditional variables. Probability in the Engineering and Informational Sciences 24: 441–455.Google Scholar
5.Belzunce, F., Franco, M. & Ruiz, J.M. (1999). On aging properties based on the residual life of k-out-of-n systems. Probability in the Engineering and Informational Sciences 13: 193–199.Google Scholar
6.Belzunce, F., Franco, M., Ruiz, J.M. & Ruiz, M.C. (2001). On partial orderings between coherent systems with different structures. Probability in the Engineering and Informational Sciences 15: 273–293.Google Scholar
7.Belzunce, F., Mercader, J.A. & Ruiz, J.M. (2005). Stochastic comparisons of generalized order statistics, Probability in the Engineering and Informational Sciences, 19, 99–120.CrossRefGoogle Scholar
8.Belzunce, F., Ruiz, J.M. & Ruiz, M.C. (2003). Multivariate properties of random vectors of order statistics. Journal of Statistical Planning and Inference 115: 413–424.CrossRefGoogle Scholar
9.Boland, P.J., Hu, T., Shaked, M. & Shanthikumar, J.G. (2002). Stochastic ordering of order statistics II. In (eds.), Dror, M., L'Ecuyer, P. & Szidarovszky, F.. Modelling uncertainty: An examination of stochastic theory, methods and applications. Boston: Kluwer, pp. 607–623.Google Scholar
10.Boland, P.J., Shaked, M. & Shanthikumar, J.G. (1998). Stochastic ordering of order statistics. In Balakrishnan, N. & Rao, C.R. (eds.), Handbook of statistics. Amsterdam: North Holland Publishing Co. Vol. 16, pp. 89–103.Google Scholar
11.Franco, M., Ruiz, J.M. & Ruiz, M.C. (2002). Stochastic orderings between spacings of generalized order statistics. Probability in the Engineering and Informational Sciences 16: 471–484.Google Scholar
12.Hu, T., Jin, W. & Khaledi, B.-E. (2007). Ordering conditional distributions of generalized order statistics. Probability in the Engineering and Informational Sciences 21: 401–417.Google Scholar
13.Hu, T. & Zhuang, W. (2005). Stochastic properties of p-spacings of generalized order statistics. Probability in the Engineering and Informational Sciences 19: 257–276.CrossRefGoogle Scholar
14.Hu, T. & Zhuang, W. (2006). A note on stochastic comparisons of generalized order statistics. Statistics and Probability Letters 72: 163–170.Google Scholar
15.Karlin, S. & Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. Journal of Multivariate Analysis 10: 467–498.Google Scholar
16.Khaledi, B.E. (2005). Some new results on stochastic comparisons between generalized order statistics. Journal of the Iranian Statistical Society 4: 35–49.Google Scholar
17.Khaledi, B.E. & Kochar, S.C. (1999). Stochastic orderings between distributions and their sampling spacings II. Statistics and Probability Letters 44: 161–166.Google Scholar
18.Khaledi, B.E. & Kochar, S.C. (2000). On dispersive ordering between order statistics in one-sample and two-sample problems. Statistics and Probability Letters 46: 257–261.CrossRefGoogle Scholar
19.Khaledi, B.E. & Kochar, S.C. (2005). Dependence orderings for generalized order statistics. Statistics and Probability Letters 73: 357–367.Google Scholar
20.Khaledi, B.E. & Shaked, M. (2007). Ordering conditional lifetimes of coherent systems. Journal of Statistical Planning and Inference 137: 1173–1184.Google Scholar
21.Kochar, S.C. & Xu, M. (2010). On residual lifetime of k-out-of-n systems with nonidentical components. Probability in the Engineering and Informational Sciences 24: 109–127.Google Scholar
22.Langberg, N., Leon, R.V. & Proschan, F. (1980). Characterization of nonparametric classes of life distributions. Annals of Probability 8: 1163–1170.Google Scholar
23.Li, X. & Chen, J. (2004). Aging properties of the residual life of k-out-of-n systems with independent but non-indentical components. Applied Stochastic Models in Business and Industry 20: 143–153.Google Scholar
24.Li, X. & Zhao, P. (2006). Some aging properties of the residual life of k-out-of-n systems. IEEE Transactions on Reliability 55: 535–541.Google Scholar
25.Li, X. & Zhao, P. (2008). Stochastic comparison on general inactivity time and general residual life of k-out-of-n systems. Communications in Statistics: Simulation and Computation 37: 1005–1019.Google Scholar
26.Li, X. & Zuo, M.J. (2002). On the behaviour of some new aging properties based upon the residual life of k-out-of-n systems. Journal of Applied Probability 39: 426–433.Google Scholar
27.Lillo, R., Nanda, A.K. & Shaked, M. (2001). Preservation of likelihood ratio stochastic orders by order statistics. Statistics and Probability Letters 51: 111–119.Google Scholar
28.Mi, J. & Shaked, M. (2002). Stochastic dominance of random variables implies the dominance of their order statistics. Journal of the Indian Statistical Association 40: 161–168.Google Scholar
29.Müller, A. & Stoyan, D. (2002). Comparison methods for stochastic models and risks. Chichester, UK: Wiley.Google Scholar
30.Navarro, J. (2008). Likelihood ratio ordering of order statistics, mixtures and systems. Journal of Statistical Planning and Inference 138: 1242–1257.Google Scholar
31.Qiu, G. & Wu, J. (2007). Some comparisons between generalized order statistics. Applied Mathematics: A Journal of Chinese Universities, Series B 22: 325–333.Google Scholar
32.Sadegh, M.K. (2008). Mean past and mean residual life functions of a parallel system with nonidentical components. Communications in Statistics: Theory and Methods 37: 1134–1145.Google Scholar
33.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer-Verlag.Google Scholar
34.Vaughan, R.J. & Venables, W.N. (1972). Permanent expressions for order statistics densities. Journal of the Royal Statistical Society: Series B 34: 308–310.Google Scholar
35.Xie, H. & Hu, T. (2008). Conditional ordering of generalized order statistics. Probability in the Engineering and Informational Sciences 22: 333–346.Google Scholar
36.Xie, H. & Hu, T. (2009). Ordering p-spacings of generalized order statistics revisited. Probability in the Engineering and Informational Sciences 23: 1–16.Google Scholar
37.Yao, J. & Hu, T. (2008). Dependence structure of a class of symmetric distributions. Communications in Statistics: Theory and Applications 37: 1338–1346.CrossRefGoogle Scholar
38.Zhao, P. & Balakrishnan, N. (2009). Stochastic comparisons of conditional generalized order statistics. Journal of Statistical Planning and Inference 139: 2920–2932.Google Scholar
39.Zhao, P., Li, X. & Balakrishnan, N. (2008). Conditional ordering of k-out-of-n systems with independent but nonidentical components. Journal of Applied Probability 45: 1113–1125.Google Scholar
40.Zhuang, W. & Hu, T. (2007). Multivariate stochastic comparisons of sequential order statistics. Probability in the Engineering and Informational Sciences 21: 47–66.Google Scholar
41.Zhuang, W., Yao, J. & Hu, T. (2010). Conditional ordering of order statistics. Journal of Multivariate Analysis 101: 640–644.Google Scholar