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COMMENTS ON THE SURVEY BY BALAKRISHNAN AND ZHAO

Published online by Cambridge University Press:  13 August 2013

Moshe Shaked
Moshe Shaked*
Affiliation:
Department of Mathematics, University of AZ, Tucson, Arizona 85721, USA E-mail: shaked@math.arizona.edu
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Extract

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 27 , Issue 4 , October 2013 , pp. 445 - 449
Copyright
Copyright © Cambridge University Press 2013 

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References

1.Argon, N.T. & Andradóttir, S. (2006). Partial pooling in tandem lines with cooperation and blocking. Queueing Systems 52: 5–30.CrossRefGoogle Scholar
2.Balakrishnan, N. & Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: A review with an emphasis on some recent developments. Probability in the Engineering and Informational Sciences 29: doi:10.1017/S0269964813000156.Google Scholar
3.Bon, J.-L. & Pǎltǎnea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5: 185–192.CrossRefGoogle ScholarPubMed
4.Genest, C., Kochar, S.C. & Xu, M. (2009). On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592.CrossRefGoogle Scholar
5.Khaledi, B.-E. & Kochar, S. (2006). Weibull distribution: Some stochastic comparisons results. Journal of Statistical Planning and Inference 136: 3121–3129.CrossRefGoogle Scholar
6.Kochar, S. & Ma, C. (1999). Dispersive ordering of convolutions of exponential random variables. Statistics and Probability Letters 43: 321–324.CrossRefGoogle Scholar
7.Kochar, S. & Rojo, J. (1996). Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions. Journal of Multivariate Analysis 59: 272–281.CrossRefGoogle Scholar
8.Kochar, S. & 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
9.Kochar, S. & Xu, M. (2010). On the right spread order of convolutions of heterogeneous exponential random variables. Journal of Multivariate Analysis 101: 165–176.CrossRefGoogle Scholar
10.Kochar, S. & Xu, M. (2011). Stochastic comparisons of spacings from heterogeneous samples. In Advances in directional and linear statistics. Wells, M.T. & SenGupta, A. (eds.), Berline: Springer, 113–129.CrossRefGoogle Scholar
11.Kochar, S.C. & Korwar, R. (1996). Stochastic orders for spacings of heterogeneous exponential random variables. Journal of Multivariate Analysis 57: 69–83.CrossRefGoogle Scholar
12.Mao, T., Hu, T. & Zhao, P. (2010). Ordering convolutions of heterogeneous exponential and geometric distributions revisited. Probability in the Engineering and Informational Sciences 24: 329–348.CrossRefGoogle Scholar
13.Pǎltǎnea, E. (2009). Stochastic ordering of sums of two independent exponential random variables. “Vasile Alecsandri” University of Bacǎu, Faculty of Sciences, Scientific Studies and Research, Series Mathematics and Informatics 19: 351–356.Google Scholar
14.Pǎltǎnea, E. (2011). Bounds for mixtures of order statistics from exponentials and applications. Journal of Multivariate Analysis 102: 896–907.CrossRefGoogle Scholar
15.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders, New York: Springer.CrossRefGoogle Scholar
16.Yu, Y. (2009). Stochastic ordering of exponential family distributions and their mixtures. Journal of Applied Probability 46: 244–254.CrossRefGoogle Scholar
17.Zhao, P. (2011). Some new results on convolutions of heterogeneous gamma random variables. Journal of Multivariate Analysis 102: 958–976.CrossRefGoogle Scholar
18.Zhao, P. & Balakrishnan, N. (2009a). Likelihood ratio ordering of convolutions of heterogeneous exponential and geometric random variables. Statistics and Probability Letters 79: 1717–1723.CrossRefGoogle Scholar
19.Zhao, P. & Balakrishnan, N. (2009b). Mean residual life order of convolutions of heterogeneous exponential random variables. Journal of Multivariate Analysis 100: 1792–1801.CrossRefGoogle Scholar
20.Zhao, P., Chan, P.S. & Ng, H.K.T. (2012). Optimal allocation of redundancies in series systems. European Journal of Operational Research 220: 673–683.CrossRefGoogle Scholar
21.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