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LIKELIHOOD RATIO AND HAZARD RATE ORDERINGS OF THE MAXIMA IN TWO MULTIPLE-OUTLIER GEOMETRIC SAMPLES

Published online by Cambridge University Press:  08 June 2012

Baojun Du ,
Peng Zhao  and
N. Balakrishnan
Baojun Du
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Peng Zhao
Affiliation:
School of Mathematical Sciences, Jiangsu Normal University, Xuzhou 221116, China
N. Balakrishnan
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, CanadaL8S 4K1 E-mail: bala@mcmaster.ca
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Abstract

In this paper, we study some stochastic comparisons of the maxima in two multiple-outlier geometric samples based on the likelihood ratio order, hazard rate order, and usual stochastic order. We establish a sufficient condition on parameter vectors for the likelihood ratio ordering to hold. For the special case when n = 2, it is proved that the p-larger order between the two parameter vectors is equivalent to the hazard rate order as well as usual stochastic order between the two maxima. Some numerical examples are presented for illustrating the established results.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 3 , July 2012 , pp. 375 - 391
Copyright
Copyright © Cambridge University Press 2012

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References

1.Andrews, D.F., Bickel, P.J., Hampel, F., Huber, P.J., Rogers, W., & Tukey, J.W. (1972). Robust estimates of location: Survey and advances. Princeton, NJ: Princeton University Press.Google Scholar
2.Balakrishnan, N. (2007). Permanents, order statistics, outliers, and robustness. Revista Matemática Complutense 20: 7–107.CrossRefGoogle Scholar
3.Balakrishnan, N. & Rao, C.R. (eds.) (1998). Handbook of statistics, vol. 16: Order statistics: Theory and methods. Amsterdam: Elsevier.Google Scholar
4.Balakrishnan, N. & Rao, C.R. (eds.) (1998). Handbook of statistics, vol. 17: Order statistics: Applications. Amsterdam: Elsevier.Google Scholar
5.Barnett, V. & Lewis, T. (1993). Outliers in statistical data, 3rd ed.Chichester, England: John Wiley & Sons.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.CrossRefGoogle Scholar
7.Bon, J.L. & Pǎltǎnea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5: 185–192.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
9.David, H.A. & Nagaraja, H.N. (2003). Order statistics, 3rd ed.Hoboken, NJ: John Wiley & Sons.CrossRefGoogle 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.CrossRefGoogle Scholar
11.Hu, T., Wang, F., & Zhu, Z. (2006). Stochastic comparisons and dependence of spacings from two samples of exponential random variables. Communications in Statistics — Theory and Methods 35: 979–988.CrossRefGoogle Scholar
12.Hu, T., Lu, Q., & Wen, S. (2008). Some new results on likelihood ratio orderings for spacings of heterogeneous exponential random variables. Communications in Statistics — Theory and Methods 37: 2506–2515.CrossRefGoogle Scholar
13.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.CrossRefGoogle Scholar
14.Khaledi, B. & Kochar, S.C. (2000). Some new results on stochastic comparisons of parallel systems. Journal of Applied Probability 37: 283–291.CrossRefGoogle Scholar
15.Khaledi, B.E. & Kochar, S. (2001). Stochastic properties of spacings in a single-outlier exponential model. Probability in the Engineering and Informational Sciences 15: 401–408.CrossRefGoogle Scholar
16.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.CrossRefGoogle Scholar
17.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.CrossRefGoogle Scholar
18.Kochar, S.C. & Xu, M. (2011). On the skewness of order statistics in the multiple-outlier models. Journal of Applied Probability 48: 271–284.CrossRefGoogle Scholar
19.Mao, T. & Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Probability in the Engineering and Informational Sciences 24: 245–262.CrossRefGoogle Scholar
20.Marshall, A.W. & Olkin, I. (1979). Inequalities: Theory of majorization and its applications. New York: Academic Press.Google Scholar
21.Pǎltǎnea, E. (2008). On the comparison in hazard rate ordering of fail-safe systems. Journal of Statistical Planning and Inference 138: 1993–1997.CrossRefGoogle Scholar
22.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
23.Proschan, F. & Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. Journal of Multivariate Analysis 6: 608–616.CrossRefGoogle Scholar
24.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer-Verlag.CrossRefGoogle Scholar
25.Tiku, M.L., Tan, W.Y., & Balakrishnan, N. (1986). Robust inference. New York: Marcel Dekker.Google Scholar
26.Wen, S., Lu, Q., & Hu, T. (2007). Likelihood ratio order of spacings from two samples of exponential distributions. Journal of Multivariate Analysis 98: 743–756.CrossRefGoogle Scholar
27.Xu, M., Li, X., Zhao, P., & Li, Z. (2007). Likelihood ratio order of m-spacings in multiple-outlier models. Communications in Statistics — Theory and Methods 36: 1507–1525.CrossRefGoogle Scholar
28.Zhao, P. & Balakrishnan, N. (2009). Characterization of MRL order of fail-safe systems with heterogeneous exponentially distributed components. Journal of Statistical and Planning Inference 139: 3027–3037.CrossRefGoogle Scholar
29.Zhao, P. & Balakrishnan, N. (2011). Some characterization results for parallel systems with two heterogeneous exponential components. Statistics 45(6): 593–604.CrossRefGoogle Scholar
30.Zhao, P. & Balakrishnan, N. (2011). Stochastic comparisons of largest order statistics from multiple-outlier exponential models. Probability in the Engineering and Informational Sciences. (in press)Google Scholar
31.Zhao, P., Li, X., & Balakrishnan, N. (2008). Conditional ordering of k-out-of-n systems with independent but non-identical components. Journal of Applied Probability 45: 1113–1125.CrossRefGoogle Scholar
32.Zhao, P., Li, X., & Balakrishnan, N. (2009). Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables. Journal of Multivariate Analysis 100: 952–962.CrossRefGoogle Scholar