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EQUIVALENT CHARACTERIZATIONS ON ORDERINGS OF ORDER STATISTICS AND SAMPLE RANGES

Published online by Cambridge University Press:  18 March 2010

Tiantian Mao  and
Taizhong Hu
Tiantian Mao
Affiliation:
Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China E-mail: mttiy@mail.ustc.edu.cn; thu@ustc.edu.cn
Taizhong Hu
Affiliation:
Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China E-mail: mttiy@mail.ustc.edu.cn; thu@ustc.edu.cn
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Abstract

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 2 , April 2010 , pp. 245 - 262
Copyright
Copyright © Cambridge University Press 2010

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References

1.Balakrishnan, N. & Rao, C.R. (1998). Handbook of Statistics 16 — Order statistics: Theory and methods. New York: Elsevier.Google Scholar
2.Balakrishnan, N. & Rao, C.R. (1998). Handbook of Statistics 17 — Order statistics: Applications. New York: Elsevier.Google Scholar
3.Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing. Silver Spring, MD: To Begin With.Google Scholar
4.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
5.Bon, J.L. & Păltănea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5: 185–192.CrossRefGoogle ScholarPubMed
6.Bon, J.L. & Păltănea, 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
7.David, H.A. & Nagaraja, H.N. (2003). Order statistics, 3rd ed.New York: Wiley.CrossRefGoogle Scholar
8.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
9.Genest, C., Kochar, S.C. & Xu, M. (2009). On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592.CrossRefGoogle Scholar
10.Hu, T. (1995). Monotone coupling and stochastic ordering of order statistics. System Science and Mathematical Science (English Series) 8: 209–214.Google Scholar
11.Khaledi, B. & Kochar, S.C. (2000). Sample range: Some stochastic comparisons results. Calcutta Statistical Association Bulletin 50: 283–291.CrossRefGoogle Scholar
12.Khaledi, B. & Kochar, S.C. (2000). Some new results on stochastic comparisons of parallel systems. Journal of Applied Probability 37: 1123–1128.CrossRefGoogle Scholar
13.Khaledi, B. & Kochar, S.C. (2002). Stochastic orderings among order statistics and sample spacings. In Misra, J.C., (ed.) Uncertainty and optimality: Probability, statistics and operations research, Singapore: World Scientific Publications, pp. 167–203.CrossRefGoogle Scholar
14.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
15.Kochar, S.C. & Xu, M. (2007). Some recent results on stochastic comparisons and dependence among order statistics in the case of PHR model. Journal of Iranian Statistical Society 6: 125–140.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 Sciences 21: 597–609.CrossRefGoogle Scholar
17.Kochar, S. & Xu, M. (2009). Comparisons of parallel systems according to the convex transform order. Journal of Applied Probability 46: 342–352.CrossRefGoogle Scholar
18.Marshall, A.W. & Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. New York: Academic Press.Google Scholar
19.Müller, A. & Stoyan, D. (2002). Comparison methods for stochastic models and risks. West Sussex, UK: Wiley.Google Scholar
20.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
21.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
22.Shaked, M. & Shanthikumar, J.G. (1998). Two variability orders. Probability in the Engineering and Informational Sciences 12: 1–23.CrossRefGoogle Scholar
23.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar
24.Zhao, P. & Li, X. (2009). Stocahstic order of sample range from heterogeneous exponential random variables. Probability in the Engineering and Informational Sciences 23: 17–29.CrossRefGoogle Scholar